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Infantile Hemangioma Meta-analysis.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Infantile Hemangioma Meta-analysis\n",
"\n",
"\n",
"> KQ 2: Among newborns, infants, and children up to 18 years of age with infantile hemangiomas who have been referred for pharmacologic intervention, what is the comparative effectiveness (benefits/harms) of corticosteroids or beta-blockers?\n",
"\n",
"Using data extracted by the systematic review, we will conduct a meta-analysis to estimate the effectiveness of several corticosteroids and beta blockers for the treatment of infantile hemangiomas. Of particular interest is the estimation of the efficacy of propanolol, a beta blocker that was used in a large number of studies in the review. To this end, we will determine the expected probability of partial-to-complete clearance for propanolol, and compare this to the same probabilities for several different comparators, as determined by their availability in the dataset.\n",
"\n",
"### Beta-binomial model\n",
"\n",
"For this model, the response variable is the number of individuals in study $j$ under intervention $k$ that achieve the clearance threshold:\n",
"\n",
"$$y_{jk} = \\sum_{i=1}^{n_{jk}} = I_i(\\text{above clearance threshold})$$\n",
"\n",
"This outcome is modeled as a binomial response:\n",
"\n",
"$$y_{jk} \\sim \\text{Bin}(n_{jk}, \\pi_{jk})$$\n",
"\n",
"where $\\pi_{jk})$ is the probability of a positive response for study $j$ under intervention $k$. To allow for heterogeneity in this probability across studies, we can specify it as a random effect:\n",
"\n",
"$$\\pi_{jk} \\sim \\text{Beta}(\\alpha, \\beta)$$\n",
"\n",
"where $\\alpha, \\beta$ are the parameters of a beta distribution (which models quantities on the [0,1] interval), resulting in a beta-binomial distribution for the outcome.\n",
"\n",
"It may be possible to incorporate covariates to improve the prediction of propanolol effectiveness. In particular, the mode of delivery (oral, intralesional, topical), dose, or the hemangioma location may be predictive of intervention effectiveness. For a vector of such covariates $x$, we can alternatively model $\\pi_{jk}$ as a logit-linear function:\n",
"\n",
"$$\\text{logit}(\\pi_{jk}) = x_j^{\\prime}\\theta_k + \\epsilon_j$$\n",
"\n",
"where $\\theta$ is a corresponding vector of regression parameters corresponding to intervention $k$, and $\\epsilon_j$ is a study-level random effect to account for the correlation of study arms.\n",
"\n",
"### Latent variable model\n",
"\n",
"However, the use of an arbitrary cutoff value as a threshold of success is an unfortunate and perhaps unsatisfactory modeling choice. There is an inherent loss of information in the dichotomization of continuous variables, and this loss is magnified here by having to discard data from studies that use a different response threshold than the adopted value (*e.g.* 75%). Since the clearance rate is a continuous measure, one can hypothesize a latent, continuous probability distribution that each study reports according to its respective quantiles: 25%, 50%, 75%, etc. If there is sufficient information, one may use a Bayesian approach to attempt to reconstruct this latent distribution, which would allow for more of the available information to be used in the meta-analytic procedure. \n",
"\n",
"Under some treatment $k$, one can consider a notional *distribution* of hemangioma clearance rates, from no effect (0) to complete clearance (1). As a matter of convenience in any particular study $j$, researchers will have chosen an arbitrary clearance threshold $c_j$, only recording whether a particular subject occupied one side or the other of this threshold. We can characterize the true, latent response distribution by estimating the parameters $\\mu_k, \\sigma_k$ via the following identity:\n",
"\n",
"$$\\pi_{jk} = 1 - I_{c_j}(\\mu_k, \\sigma_k)$$\n",
"\n",
"where $I_x(a,b)$ is the cumulative distribution function of a logit-transformed normal distribution under parameters $\\mu, \\sigma$. The resulting probability is the same as specified above, and can be used in the same binomial likelihood:\n",
"\n",
"$$y_{jk} \\sim \\text{Bin}(n_{jk}, \\pi_{jk})$$\n",
"\n",
"### Comparative effectiveness\n",
"\n",
"Irrespective of which model form is employed, the comparative effectiveness of any two pharmacologic interventions $k=1,2$ can be assessed directly by the difference in their respective clearance threshold probabilities:\n",
"\n",
"$$d_{12} = \\pi_1 - \\pi_2$$\n",
"\n",
"Using a Bayesian framework, we can extract associated 95% posterior credible intervals for this difference."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import pandas as pd\n",
"import pymc as pm\n",
"import seaborn as sb\n",
"import pylab as pl\n",
"\n",
"from scipy.stats import norm"
]
},
{
"cell_type": "code",
"execution_count": 132,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sb.set_style(\"white\")"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# Set random number seed\n",
"np.random.seed(42)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"kq2_outcomes = pd.read_excel('IH_effectiveness_data_Extraction_comparative studies_revised_03242015.xlsx', \n",
" sheetname='KQ2_outcome data')\n",
"\n",
"# Individual patient data for RefID 112\n",
"kq2_ipd = pd.read_excel('IH_effectiveness_data_Extraction_comparative studies_revised_03242015.xlsx', \n",
" sheetname='KQ2_IPD')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Individual patient data for REFID 112 (Qiu 2013). Individual with multiple treatments per lesion was deleted; data from multiple lesions on the same individual were averaged."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"unique_outcomes = kq2_outcomes['Outcome'].unique()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array(['propranolol', 'atenolol', 'Timolol maleate 0.5% gel', 'placebo',\n",
" 'triamcilone', 'prednisolone', 'control', 'control - no treatment',\n",
" 'timolol', 'Imiquimod', 'bleomycin', 'prednisone',\n",
" 'corticosteroids', 'No treatment - observation', 'triamcinolone',\n",
" 'nadolol', 'Methylprednisolone'], dtype=object)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unique_interventions = kq2_outcomes['Arm'].unique()\n",
"\n",
"unique_interventions"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"intervention_categories = pd.read_csv('interventions.csv', index_col=0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Remove unusable interventions"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"dropped_interventions = intervention_categories[intervention_categories.Label=='DROP'].index.values\n",
"kq2_outcomes = kq2_outcomes[~kq2_outcomes.Arm.isin(dropped_interventions)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Recode interventions into collapsed category set"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"intervention_lookup = intervention_categories[intervention_categories.Label!='DROP'].to_dict()['Label']"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"{'Control (not treated with propranolol)': 'control',\n",
" 'Imiquimod': 'imiquimod',\n",
" 'Methylprednisolone': 'methylprednisolone',\n",
" 'No treatment - observation': 'control',\n",
" 'Timolol maleate 0.5% gel': 'timolol',\n",
" 'atenolol': 'atenolol',\n",
" 'control': 'control',\n",
" 'control - no treatment': 'control',\n",
" 'corticosteroids': 'oral steroid',\n",
" 'methyprednisolone': 'methylprednisolone',\n",
" 'nadolol': 'nadolol',\n",
" 'nonpropranolol': 'control',\n",
" 'placebo': 'control',\n",
" 'prednisolone': 'oral steroid',\n",
" 'prednisone': 'oral steroid',\n",
" 'propranolol': 'propranolol',\n",
" 'timolol': 'timolol',\n",
" 'triamcilone': 'triamcilone',\n",
" 'triamcilone acetonide': 'triamcilone',\n",
" 'triamcinolone': 'triamcilone'}"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"intervention_lookup"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"propranolol 34\n",
"control 15\n",
"triamcilone 9\n",
"timolol 9\n",
"oral steroid 8\n",
"atenolol 2\n",
"nadolol 1\n",
"imiquimod 1\n",
"methylprednisolone 1\n",
"dtype: int64"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes['intervention'] = kq2_outcomes.Arm.replace(intervention_lookup)\n",
"kq2_outcomes['intervention'].value_counts()"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"8"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unique_interventions = kq2_outcomes.intervention.unique()\n",
"n_interventions = len(unique_interventions) - 1\n",
"n_interventions"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array(['propranolol', 'atenolol', 'timolol', 'control', 'triamcilone',\n",
" 'oral steroid', 'imiquimod', 'nadolol', 'methylprednisolone'], dtype=object)"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unique_interventions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Define indices to intervention elements"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"PROPRANOLOL = 0\n",
"ATENOLOL = 1\n",
"TIMOLOL = 2\n",
"TRIAMCILONE = 3\n",
"ORAL_STEROID = 4\n",
"IMIQUIMOD = 5\n",
"NADOLOL = 6\n",
"METHYLPREDNISOLONE = 7"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Number of unique studies"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"17"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unique_studies = kq2_outcomes.REFID.unique()\n",
"n_studies = len(unique_studies)\n",
"n_studies"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 5, 110, 1263, 43, 13, 445, 3740, 72, 112, 309, 321,\n",
" 402, 438, 3451, 3522, 3723, 1180])"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unique_studies"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There does not appear to be sufficient variation in dosage to estimate its effect for any agent."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"Dosage form Dosage amount intervention \n",
"intralesional mg/kg triamcilone 1 to 5 3\n",
" mg/ml propranolol 1 10\n",
" triamcilone 40 6\n",
"intravenous mg/kg methylprednisolone 30 1\n",
"oral mg/kg atenolol 1 2\n",
" nadolol up to 4.0 1\n",
" oral steroid 2.0 5\n",
" 4.0 2\n",
" 2.8 1\n",
" propranolol 2.0 17\n",
" 2 to 3 1\n",
" 3 1\n",
" 2.7 1\n",
"topical BID propranolol 0.01 4\n",
" timolol 0.0025 3\n",
" Drop control 1 1\n",
" timolol 1 1\n",
" NR timolol NR 4\n",
"dtype: int64"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes.groupby(['Dosage form','Dosage amount','intervention'])['Dosage'].value_counts()"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"intervention Dosage form \n",
"atenolol oral 2\n",
"control topical 1\n",
"imiquimod topical 1\n",
"methylprednisolone intravenous 1\n",
"nadolol oral 1\n",
"oral steroid oral 8\n",
"propranolol intralesional 10\n",
" oral 20\n",
" topical 4\n",
"timolol topical 9\n",
"triamcilone intralesional 9\n",
"Name: REFID, dtype: int64"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes.groupby(['intervention', 'Dosage form']).REFID.count()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Model specification\n",
"\n",
"Study random effect. Might want to use a robust distribution, like Cauchy or t."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"σ_ϵ = pm.Uniform('σ_ϵ', 0, 1000, value=10)\n",
"τ_ϵ = σ_ϵ ** -2\n",
"\n",
"ϵ = pm.Normal('ϵ', 0, τ_ϵ, value=np.zeros(n_studies))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Parameters for each intervention"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"μ = pm.Normal('μ', 0, 0.001, value=-2)\n",
"θ = pm.Normal('θ', 0, 0.001, value=np.zeros(n_interventions))\n",
"λ = pm.Exponential('λ', 0.1, value=0.5)\n",
"σ = pm.HalfNormal('σ', λ, value=np.ones(n_interventions+1))\n",
"τ = σ ** -2"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(8,)"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"θ.value.shape"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(9,)"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"σ.value.shape"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Alternate mode of delivery effects for propanolol (oral is baseline)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"ϕ = pm.Normal('ϕ', 0, 0.001, value=0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Abarrzua-Araya 2014 (RefID 5)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"subset_cols = ['REFID', 'Arm', 'Dosage', 'Dosage amount', 'Dosage form', 'N at follow-up', 'Outcome', \n",
" 'Outcome data/result', 'Outcome data', 'Point estimate', 'sd', 'Lower range', 'Upper range', 'Outcome data/result N']"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td> 5</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> Complete response</td>\n",
" <td> 6 (60)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td> 5</td>\n",
" <td> atenolol</td>\n",
" <td> 1</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 13</td>\n",
" <td> Complete response</td>\n",
" <td> 7 (53.8)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 7</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td> 5</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> Partial response</td>\n",
" <td> 4 (40)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 99</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td> 5</td>\n",
" <td> atenolol</td>\n",
" <td> 1</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 13</td>\n",
" <td> Partial response</td>\n",
" <td> 6 (46.1)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 99</td>\n",
" <td> 6</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"0 5 propranolol 2 mg/kg oral 10 \n",
"1 5 atenolol 1 mg/kg oral 13 \n",
"2 5 propranolol 2 mg/kg oral 10 \n",
"3 5 atenolol 1 mg/kg oral 13 \n",
"\n",
" Outcome Outcome data/result Outcome data Point estimate sd \\\n",
"0 Complete response 6 (60) range NaN NaN \n",
"1 Complete response 7 (53.8) range NaN NaN \n",
"2 Partial response 4 (40) range NaN NaN \n",
"3 Partial response 6 (46.1) range NaN NaN \n",
"\n",
" Lower range Upper range Outcome data/result N \n",
"0 99 100 6 \n",
"1 99 100 7 \n",
"2 1 99 4 \n",
"3 1 99 6 "
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==5][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"refid5 = np.where(unique_studies==5)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_5(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for propanolol in RefID 5\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid5]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_5(μ=mu_propranolol_5, τ=τ):\n",
" # Response category probabilities for propanolol in RefID 5\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.99), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for propanolol in RefID 5\n",
"y_propanolol_5 = pm.Multinomial('y_propanolol_5', 10, p_propranolol_5, value=[0, 4, 6], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_atenolol_5(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for atenolol in RefID 5\n",
" return μ + θ[ATENOLOL] + ϵ[refid5]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_atenolol_5(μ=mu_atenolol_5, τ=τ): \n",
" # Response category probabilities for atenolol in RefID 5\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[ATENOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.99), μ, τ[ATENOLOL]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for atenolol in RefID 5\n",
"y_atenolol_5 = pm.Multinomial('y_atenolol_5', 13, p_atenolol_5, value=[0, 6, 7], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Chan 2013 (RefID 110)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>4</th>\n",
" <td> 110</td>\n",
" <td> Timolol maleate 0.5% gel</td>\n",
" <td> 1</td>\n",
" <td> Drop</td>\n",
" <td> topical</td>\n",
" <td> 15</td>\n",
" <td> Volume reduced by ≥ 5%</td>\n",
" <td> 60</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 5</td>\n",
" <td> 100</td>\n",
" <td> 9</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td> 110</td>\n",
" <td> placebo</td>\n",
" <td> 1</td>\n",
" <td> Drop</td>\n",
" <td> topical</td>\n",
" <td> 18</td>\n",
" <td> Volume reduced by ≥ 5%</td>\n",
" <td> 11</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 5</td>\n",
" <td> 100</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form \\\n",
"4 110 Timolol maleate 0.5% gel 1 Drop topical \n",
"5 110 placebo 1 Drop topical \n",
"\n",
" N at follow-up Outcome Outcome data/result Outcome data \\\n",
"4 15 Volume reduced by ≥ 5% 60 range \n",
"5 18 Volume reduced by ≥ 5% 11 range \n",
"\n",
" Point estimate sd Lower range Upper range Outcome data/result N \n",
"4 NaN NaN 5 100 9 \n",
"5 NaN NaN 5 100 2 "
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==110][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid110 = np.where(unique_studies==110)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_timolol_110(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for timolol in RefID 110\n",
" return μ + θ[TIMOLOL] + ϵ[refid110]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_timolol_110(μ=mu_timolol_110, τ=τ):\n",
" # Response category probabilities for timolol in RefID 110\n",
" return 1 - norm.cdf(pm.logit(0.05), μ, τ[TIMOLOL]**-1)\n",
"\n",
"# Binomial likelihood for propanolol in RefID 5\n",
"y_timolol_110 = pm.Binomial('y_timolol_110', 15, p_timolol_110, value=9, observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_110(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for control in RefID 110\n",
" return μ + ϵ[refid110]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_110(μ=mu_control_110, τ=τ):\n",
" # Response category probabilities for control in RefID 110\n",
" return 1 - norm.cdf(pm.logit(0.05), μ, τ[-1]**-1)\n",
"\n",
"# Binomial likelihood for control in RefID 5\n",
"y_control_110 = pm.Binomial('y_control_110', 18, p_control_110, value=2, observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Jalil 2006 (RefID 1263)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>6 </th>\n",
" <td> 1263</td>\n",
" <td> triamcilone</td>\n",
" <td> 1 to 5</td>\n",
" <td> mg/kg</td>\n",
" <td> intralesional</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &gt; 50%</td>\n",
" <td> 11 (44)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 11</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7 </th>\n",
" <td> 1263</td>\n",
" <td> prednisolone</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &gt; 50%</td>\n",
" <td> 8 (32)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 8</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8 </th>\n",
" <td> 1263</td>\n",
" <td> control</td>\n",
" <td> no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &gt; 50%</td>\n",
" <td> 0</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>9 </th>\n",
" <td> 1263</td>\n",
" <td> triamcilone</td>\n",
" <td> 1 to 5</td>\n",
" <td> mg/kg</td>\n",
" <td> intralesional</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &lt; 50%</td>\n",
" <td> 8 (32)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> 50</td>\n",
" <td> 8</td>\n",
" </tr>\n",
" <tr>\n",
" <th>10</th>\n",
" <td> 1263</td>\n",
" <td> prednisolone</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &lt; 50%</td>\n",
" <td> 11 (44)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> 50</td>\n",
" <td> 11</td>\n",
" </tr>\n",
" <tr>\n",
" <th>11</th>\n",
" <td> 1263</td>\n",
" <td> control</td>\n",
" <td> no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction &lt; 50%</td>\n",
" <td> 1 (4)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> 50</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>12</th>\n",
" <td> 1263</td>\n",
" <td> triamcilone</td>\n",
" <td> 1 to 5</td>\n",
" <td> mg/kg</td>\n",
" <td> intralesional</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction little or none</td>\n",
" <td> 6 (24)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 10</td>\n",
" <td> 6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>13</th>\n",
" <td> 1263</td>\n",
" <td> prednisolone</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction little or none</td>\n",
" <td> 6 (24)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 10</td>\n",
" <td> 6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>14</th>\n",
" <td> 1263</td>\n",
" <td> control</td>\n",
" <td> no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 25</td>\n",
" <td> Lesion size reduction little or none</td>\n",
" <td> 19 (76)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 10</td>\n",
" <td> 19</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form \\\n",
"6 1263 triamcilone 1 to 5 mg/kg intralesional \n",
"7 1263 prednisolone 2 mg/kg oral \n",
"8 1263 control no treatment NaN NaN \n",
"9 1263 triamcilone 1 to 5 mg/kg intralesional \n",
"10 1263 prednisolone 2 mg/kg oral \n",
"11 1263 control no treatment NaN NaN \n",
"12 1263 triamcilone 1 to 5 mg/kg intralesional \n",
"13 1263 prednisolone 2 mg/kg oral \n",
"14 1263 control no treatment NaN NaN \n",
"\n",
" N at follow-up Outcome Outcome data/result \\\n",
"6 25 Lesion size reduction > 50% 11 (44) \n",
"7 25 Lesion size reduction > 50% 8 (32) \n",
"8 25 Lesion size reduction > 50% 0 \n",
"9 25 Lesion size reduction < 50% 8 (32) \n",
"10 25 Lesion size reduction < 50% 11 (44) \n",
"11 25 Lesion size reduction < 50% 1 (4) \n",
"12 25 Lesion size reduction little or none 6 (24) \n",
"13 25 Lesion size reduction little or none 6 (24) \n",
"14 25 Lesion size reduction little or none 19 (76) \n",
"\n",
" Outcome data Point estimate sd Lower range Upper range \\\n",
"6 range NaN NaN 50 100 \n",
"7 range NaN NaN 50 100 \n",
"8 range NaN NaN 50 100 \n",
"9 range NaN NaN 10 50 \n",
"10 range NaN NaN 10 50 \n",
"11 range NaN NaN 10 50 \n",
"12 range NaN NaN 0 10 \n",
"13 range NaN NaN 0 10 \n",
"14 range NaN NaN 0 10 \n",
"\n",
" Outcome data/result N \n",
"6 11 \n",
"7 8 \n",
"8 0 \n",
"9 8 \n",
"10 11 \n",
"11 1 \n",
"12 6 \n",
"13 6 \n",
"14 19 "
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==1263][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid1263 = np.where(unique_studies==1263)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_triamcilone_1263(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for triamcilone in RefID 1263\n",
" return μ + θ[TRIAMCILONE] + ϵ[refid1263]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_triamcilone_1263(μ=mu_triamcilone_1263, τ=τ):\n",
" # Response category probabilities for triamcilone in RefID 1263\n",
" c1 = norm.cdf(pm.logit(0.1), μ, τ[TRIAMCILONE]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[TRIAMCILONE]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for triamcilone in RefID 1263\n",
"y_triamcilone_1263 = pm.Multinomial('y_triamcilone_1263', 25, p_triamcilone_1263, value=[6, 8, 11], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_prednisolone_1263(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for prednisolone in RefID 1263\n",
" return μ + θ[ORAL_STEROID] + ϵ[refid1263]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_prednisolone_1263(μ=mu_triamcilone_1263, τ=τ):\n",
" # Response category probabilities for prednisolone in RefID 1263\n",
" c1 = norm.cdf(pm.logit(0.1), μ, τ[ORAL_STEROID]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[ORAL_STEROID]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for prednisolone in RefID 1263\n",
"y_prednisolone_1263 = pm.Multinomial('y_prednisolone_1263', 25, p_prednisolone_1263, value=[6, 11, 8], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_1263(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for control in RefID 1263\n",
" return μ + ϵ[refid1263]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_1263(μ=mu_control_1263, τ=τ):\n",
" # Response category probabilities for prednisolone in RefID 1263\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[-1]**-1)\n",
" c2 = norm.cdf(pm.logit(0.1), μ, τ[-1]**-1)\n",
" c3 = norm.cdf(pm.logit(0.5), μ, τ[-1]**-1)\n",
" return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# Multinomial likelihood for control in RefID 1263\n",
"y_control_1263 = pm.Multinomial('y_control_1263', 25, p_control_1263, value=[5, 19, 1, 0], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zahar 2013 (RefID 43)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>15</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 15</td>\n",
" <td> Excellent response (complete resolution)</td>\n",
" <td> 9 (60)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 9</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 0.01</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 15</td>\n",
" <td> Excellent response (complete resolution)</td>\n",
" <td> 3 (20)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>17</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 15</td>\n",
" <td> Excellent response (complete resolution)</td>\n",
" <td> 2 (13.3)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>18</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 15</td>\n",
" <td> Good response (≥50% reduction in size of IH)</td>\n",
" <td> 2 (13.3)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 99</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>19</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 0.01</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 15</td>\n",
" <td> Good response (≥50% reduction in size of IH)</td>\n",
" <td> 5 (33.3)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 99</td>\n",
" <td> 5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>20</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 15</td>\n",
" <td> Good response (≥50% reduction in size of IH)</td>\n",
" <td> 3 (20)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 99</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>21</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 15</td>\n",
" <td> Fair response (&lt;50% reduction in size of IH)</td>\n",
" <td> 1 (6.7)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 50</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>22</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 0.01</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 15</td>\n",
" <td> Fair response (&lt;50% reduction in size of IH)</td>\n",
" <td> 2 (13.3)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 50</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>23</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 15</td>\n",
" <td> Fair response (&lt;50% reduction in size of IH)</td>\n",
" <td> 3 (20)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 50</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>24</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 15</td>\n",
" <td> Poor response (no response, worsening of IH)</td>\n",
" <td> 3 (20)</td>\n",
" <td> scalar</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>25</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 0.01</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 15</td>\n",
" <td> Poor response (no response, worsening of IH)</td>\n",
" <td> 5 (33.3)</td>\n",
" <td> scalar</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>26</th>\n",
" <td> 43</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 15</td>\n",
" <td> Poor response (no response, worsening of IH)</td>\n",
" <td> 7 (46.7)</td>\n",
" <td> scalar</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 7</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"15 43 propranolol 2 mg/kg oral 15 \n",
"16 43 propranolol 0.01 BID topical 15 \n",
"17 43 propranolol 1 mg/ml intralesional 15 \n",
"18 43 propranolol 2 mg/kg oral 15 \n",
"19 43 propranolol 0.01 BID topical 15 \n",
"20 43 propranolol 1 mg/ml intralesional 15 \n",
"21 43 propranolol 2 mg/kg oral 15 \n",
"22 43 propranolol 0.01 BID topical 15 \n",
"23 43 propranolol 1 mg/ml intralesional 15 \n",
"24 43 propranolol 2 mg/kg oral 15 \n",
"25 43 propranolol 0.01 BID topical 15 \n",
"26 43 propranolol 1 mg/ml intralesional 15 \n",
"\n",
" Outcome Outcome data/result \\\n",
"15 Excellent response (complete resolution) 9 (60) \n",
"16 Excellent response (complete resolution) 3 (20) \n",
"17 Excellent response (complete resolution) 2 (13.3) \n",
"18 Good response (≥50% reduction in size of IH) 2 (13.3) \n",
"19 Good response (≥50% reduction in size of IH) 5 (33.3) \n",
"20 Good response (≥50% reduction in size of IH) 3 (20) \n",
"21 Fair response (<50% reduction in size of IH) 1 (6.7) \n",
"22 Fair response (<50% reduction in size of IH) 2 (13.3) \n",
"23 Fair response (<50% reduction in size of IH) 3 (20) \n",
"24 Poor response (no response, worsening of IH) 3 (20) \n",
"25 Poor response (no response, worsening of IH) 5 (33.3) \n",
"26 Poor response (no response, worsening of IH) 7 (46.7) \n",
"\n",
" Outcome data Point estimate sd Lower range Upper range \\\n",
"15 range NaN NaN 99 100 \n",
"16 range NaN NaN 99 100 \n",
"17 range NaN NaN 99 100 \n",
"18 range NaN NaN 50 99 \n",
"19 range NaN NaN 50 99 \n",
"20 range NaN NaN 50 99 \n",
"21 range NaN NaN 1 50 \n",
"22 range NaN NaN 1 50 \n",
"23 range NaN NaN 1 50 \n",
"24 scalar NaN NaN 0 1 \n",
"25 scalar NaN NaN 0 1 \n",
"26 scalar NaN NaN 0 1 \n",
"\n",
" Outcome data/result N \n",
"15 9 \n",
"16 3 \n",
"17 2 \n",
"18 2 \n",
"19 5 \n",
"20 3 \n",
"21 1 \n",
"22 2 \n",
"23 3 \n",
"24 3 \n",
"25 5 \n",
"26 7 "
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==43][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid43 = np.where(unique_studies==43)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_oral_43(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 43\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid43]\n",
"\n",
"@pm.deterministic\n",
"def mu_propranolol_oral_43(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 43\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid43]\n",
"@pm.deterministic\n",
"def p_propranolol_oral_43(μ=mu_propranolol_oral_43, τ=τ):\n",
" # Response category probabilities for oral propanolol in RefID 43\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.50), μ, τ[PROPRANOLOL]**-1)\n",
" c3 = norm.cdf(pm.logit(0.99), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# Multinomial likelihood for oral propanolol in RefID 43\n",
"y_propanolol_oral_43 = pm.Multinomial('y_propanolol_oral_43', 15, p_propranolol_oral_43, value=[3, 1, 2, 9], \n",
" observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# @pm.deterministic\n",
"# def mu_propranolol_topical_43(μ=μ, θ=θ, ϕ=ϕ, ϵ=ϵ):\n",
"# # Mean response on the logit scale for topical propanolol in RefID 43\n",
"# return μ + θ[PROPRANOLOL] + ϕ + ϵ[refid43]\n",
"\n",
"\n",
"# @pm.deterministic\n",
"# def p_propranolol_topical_43(μ=mu_propranolol_topical_43, τ=τ):\n",
"# # Response category probabilities for topical propanolol in RefID 43\n",
"# c1 = norm.cdf(pm.logit(0.01), μ, τ**-1) #[PROPRANOLOL]**-1)\n",
"# c2 = norm.cdf(pm.logit(0.50), μ, τ**-1) #[PROPRANOLOL]**-1)\n",
"# c3 = norm.cdf(pm.logit(0.99), μ, τ**-1) #[PROPRANOLOL]**-1)\n",
"# return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# # Multinomial likelihood for topical propanolol in RefID 43\n",
"# y_propanolol_topical_43 = pm.Multinomial('y_propanolol_topical_43', 15, p_propranolol_topical_43, value=[5, 2, 5, 3], \n",
"# observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_intralesional_43(μ=μ, θ=θ, ϕ=ϕ, ϵ=ϵ):\n",
" # Mean response on the logit scale for intralesional propanolol in RefID 43\n",
" return μ + θ[PROPRANOLOL] + ϕ + ϵ[refid43]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_intralesional_43(μ=mu_propranolol_intralesional_43, τ=τ):\n",
" # Response category probabilities for intralesional propanolol in RefID 43\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.50), μ, τ[PROPRANOLOL]**-1)\n",
" c3 = norm.cdf(pm.logit(0.99), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# Multinomial likelihood for intralesional propanolol in RefID 43\n",
"y_propanolol_intralesional_43 = pm.Multinomial('y_propanolol_intralesional_43', 15, p_propranolol_intralesional_43, \n",
" value=[7, 3, 3, 2], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Bauman 2014 (RefID 13)\n",
"\n",
"This data was given as a mean change on TSA, with 95% confidence interval. We will use a normal factor potential on augmented data based on the estimated mean, and the back-calculated standard deviation."
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>27</th>\n",
" <td> 13</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 9</td>\n",
" <td> Change in TSA, square millimeters</td>\n",
" <td> 0.64 [0.44-0.83]</td>\n",
" <td> estimate</td>\n",
" <td> 0.64</td>\n",
" <td> Back calculate</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>28</th>\n",
" <td> 13</td>\n",
" <td> prednisolone</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 6</td>\n",
" <td> Change in TSA, square millimeters</td>\n",
" <td> 0.41 [0.10-0.72]</td>\n",
" <td> estimate</td>\n",
" <td> 0.41</td>\n",
" <td> Back calculate</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"27 13 propranolol 2 mg/kg oral 9 \n",
"28 13 prednisolone 2 mg/kg oral 6 \n",
"\n",
" Outcome Outcome data/result Outcome data \\\n",
"27 Change in TSA, square millimeters 0.64 [0.44-0.83] estimate \n",
"28 Change in TSA, square millimeters 0.41 [0.10-0.72] estimate \n",
"\n",
" Point estimate sd Lower range Upper range \\\n",
"27 0.64 Back calculate NaN NaN \n",
"28 0.41 Back calculate NaN NaN \n",
"\n",
" Outcome data/result N \n",
"27 NaN \n",
"28 NaN "
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==13][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid13 = np.where(unique_studies==13)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"_propranolol_sd = 9 * (0.83 - 0.44) / 3.92\n",
"_prednisolone_sd = 6 * (0.72 - 0.10) / 3.92"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"propranolol_13_var = (_propranolol_sd**2) / (0.64 * (1-0.64))**2\n",
"prednisolone_13_var = (_prednisolone_sd**2) / (0.41 * (1-0.41))**2"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_13(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 13\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid13]\n",
"\n",
"propranolol_obs_13 = pm.Normal('propranolol_obs_13', mu_propranolol_13, propranolol_13_var**-1, \n",
" value=pm.logit(0.64), observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_prednisolone_13(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for prednisolone in RefID 43\n",
" return μ + θ[ORAL_STEROID] + ϵ[refid13]\n",
"\n",
"prednisolone_obs_13 = pm.Normal('prednisolone_obs_13', mu_prednisolone_13, prednisolone_13_var**-1, \n",
" value=pm.logit(0.41), observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Hogeling 2011 (RefID 445)"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>29</th>\n",
" <td> 445</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 18</td>\n",
" <td> Change in volume</td>\n",
" <td> -60</td>\n",
" <td> estimate</td>\n",
" <td> 0.600</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> 18</td>\n",
" </tr>\n",
" <tr>\n",
" <th>30</th>\n",
" <td> 445</td>\n",
" <td> placebo</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 15</td>\n",
" <td> Change in volume</td>\n",
" <td>-14.1</td>\n",
" <td> estimate</td>\n",
" <td> 0.147</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> 15</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"29 445 propranolol 2 mg/kg oral 18 \n",
"30 445 placebo NaN NaN NaN 15 \n",
"\n",
" Outcome Outcome data/result Outcome data Point estimate sd \\\n",
"29 Change in volume -60 estimate 0.600 NaN \n",
"30 Change in volume -14.1 estimate 0.147 NaN \n",
"\n",
" Lower range Upper range Outcome data/result N \n",
"29 NaN NaN 18 \n",
"30 NaN NaN 15 "
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==445][subset_cols]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Hogeling et al did not report a measure of uncertainty with the effect sizes, therefore this study must be excluded from the meta-analysis."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Leaute-Labreze 2015 (RefID 3740)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>31</th>\n",
" <td> 3740</td>\n",
" <td> propranolol</td>\n",
" <td> 3</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 101</td>\n",
" <td> Complete or nearly complete resolution</td>\n",
" <td> 61 (60)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 90</td>\n",
" <td> 100</td>\n",
" <td> 61</td>\n",
" </tr>\n",
" <tr>\n",
" <th>32</th>\n",
" <td> 3740</td>\n",
" <td> placebo</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 25</td>\n",
" <td> Complete or nearly complete resolution</td>\n",
" <td> 2 (4)</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 90</td>\n",
" <td> 100</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"31 3740 propranolol 3 mg/kg oral 101 \n",
"32 3740 placebo NaN NaN NaN 25 \n",
"\n",
" Outcome Outcome data/result Outcome data \\\n",
"31 Complete or nearly complete resolution 61 (60) range \n",
"32 Complete or nearly complete resolution 2 (4) range \n",
"\n",
" Point estimate sd Lower range Upper range Outcome data/result N \n",
"31 NaN NaN 90 100 61 \n",
"32 NaN NaN 90 100 2 "
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==3740][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid3740 = np.where(unique_studies==3740)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propanolol_3740(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for propanolol in RefID 3740\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid3740]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propanolol_3740(μ=mu_propanolol_3740, τ=τ):\n",
" # Response category probabilities for propanolol in RefID 3740\n",
" return 1 - norm.cdf(pm.logit(0.9), μ, τ[PROPRANOLOL]**-1)\n",
"\n",
"# Binomial likelihood for propanolol in RefID 5\n",
"y_propanolol_3740 = pm.Binomial('y_propanolol_3740', 101, p_propanolol_3740, value=61, \n",
" observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_3740(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for controls in RefID 3740\n",
" return μ + ϵ[refid3740]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_3740(μ=mu_control_3740, τ=τ):\n",
" # Response category probabilities for controls in RefID 3740\n",
" return 1 - norm.cdf(pm.logit(0.9), μ, τ[-1]**-1)\n",
"\n",
"# Binomial likelihood for propanolol in RefID 5\n",
"y_control_3740 = pm.Binomial('y_control_3740', 25, p_control_3740, value=2, \n",
" observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Sondhi 2013 (RefID 72)\n",
"\n",
"The paper claims `n=14` for controls, but only reports 12 outcomes."
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>33</th>\n",
" <td> 72</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 31</td>\n",
" <td> Significant involution (&gt;50%)</td>\n",
" <td> 28</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 28</td>\n",
" </tr>\n",
" <tr>\n",
" <th>34</th>\n",
" <td> 72</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 31</td>\n",
" <td> Some involution (11%-50%)</td>\n",
" <td> 0</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 11</td>\n",
" <td> 50</td>\n",
" <td> 0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>35</th>\n",
" <td> 72</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 31</td>\n",
" <td> No involution (≤10%)</td>\n",
" <td> 3</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 10</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>36</th>\n",
" <td> 72</td>\n",
" <td> control - no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 14</td>\n",
" <td> Significant involution (&gt;50%)</td>\n",
" <td> 4</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>37</th>\n",
" <td> 72</td>\n",
" <td> control - no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 14</td>\n",
" <td> Some involution (11%-50%)</td>\n",
" <td> 2</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 11</td>\n",
" <td> 50</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>38</th>\n",
" <td> 72</td>\n",
" <td> control - no treatment</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 14</td>\n",
" <td> No involution (≤10%)</td>\n",
" <td> 6</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 10</td>\n",
" <td> 6</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form \\\n",
"33 72 propranolol 2 mg/kg oral \n",
"34 72 propranolol 2 mg/kg oral \n",
"35 72 propranolol 2 mg/kg oral \n",
"36 72 control - no treatment NaN NaN NaN \n",
"37 72 control - no treatment NaN NaN NaN \n",
"38 72 control - no treatment NaN NaN NaN \n",
"\n",
" N at follow-up Outcome Outcome data/result \\\n",
"33 31 Significant involution (>50%) 28 \n",
"34 31 Some involution (11%-50%) 0 \n",
"35 31 No involution (≤10%) 3 \n",
"36 14 Significant involution (>50%) 4 \n",
"37 14 Some involution (11%-50%) 2 \n",
"38 14 No involution (≤10%) 6 \n",
"\n",
" Outcome data Point estimate sd Lower range Upper range \\\n",
"33 range NaN NaN 50 100 \n",
"34 range NaN NaN 11 50 \n",
"35 range NaN NaN 0 10 \n",
"36 range NaN NaN 50 100 \n",
"37 range NaN NaN 11 50 \n",
"38 range NaN NaN 0 10 \n",
"\n",
" Outcome data/result N \n",
"33 28 \n",
"34 0 \n",
"35 3 \n",
"36 4 \n",
"37 2 \n",
"38 6 "
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==72][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid72 = np.where(unique_studies==72)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_72(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for propanolol in RefID 72\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid72]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_72(μ=mu_propranolol_72, τ=τ):\n",
" # Response category probabilities for propanolol in RefID 72\n",
" c1 = norm.cdf(pm.logit(0.1), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for propanolol in RefID 5\n",
"y_propanolol_72 = pm.Multinomial('y_propanolol_72', 31, p_propranolol_72, value=[3, 0, 28], \n",
" observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_72(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for controls in RefID 72\n",
" return μ + ϵ[refid72]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_72(μ=mu_control_72, τ=τ):\n",
" # Response category probabilities for controls in RefID 72\n",
" c1 = norm.cdf(pm.logit(0.1), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, 1-c2 \n",
"\n",
"# Multinomial likelihood for propanolol in RefID 5\n",
"y_control_72 = pm.Multinomial('y_control_72', 12, p_control_72, value=[6, 2, 4], \n",
" observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Qiu 2013 (RefID 112)\n",
"\n",
"Qiu et al. report individual patient data, which we incorporate directly into the meta-analysis."
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>39</th>\n",
" <td> 112</td>\n",
" <td> timolol</td>\n",
" <td> 0.005</td>\n",
" <td> NaN</td>\n",
" <td> topical</td>\n",
" <td> 20</td>\n",
" <td> Visual analog scale</td>\n",
" <td> NR</td>\n",
" <td> IPD</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NR</td>\n",
" </tr>\n",
" <tr>\n",
" <th>40</th>\n",
" <td> 112</td>\n",
" <td> Imiquimod</td>\n",
" <td> 0.05</td>\n",
" <td> NaN</td>\n",
" <td> topical</td>\n",
" <td> 20</td>\n",
" <td> Visual analog scale</td>\n",
" <td> NR</td>\n",
" <td> IPD</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NR</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"39 112 timolol 0.005 NaN topical 20 \n",
"40 112 Imiquimod 0.05 NaN topical 20 \n",
"\n",
" Outcome Outcome data/result Outcome data Point estimate sd \\\n",
"39 Visual analog scale NR IPD NaN NaN \n",
"40 Visual analog scale NR IPD NaN NaN \n",
"\n",
" Lower range Upper range Outcome data/result N \n",
"39 NaN NaN NR \n",
"40 NaN NaN NR "
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==112][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid112 = np.where(unique_studies==112)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"obs_imiquimod_112 = kq2_ipd[kq2_ipd.Arm=='imiquimod'].Outcome.values/100.\n",
"obs_timolol_112 = kq2_ipd[kq2_ipd.Arm=='timolol'].Outcome.values/100."
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"77.5"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_ipd[kq2_ipd.Arm=='imiquimod'].Outcome.median()"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_timolol_112(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for timolol in RefID 112\n",
" return μ + θ[TIMOLOL] + ϵ[refid112]\n",
"\n",
"\n",
"y_timolol_112 = pm.Normal('y_timolol_112', mu_timolol_112, τ[TIMOLOL]**-1, \n",
" value=pm.logit(obs_timolol_112 - 1e-6), observed=True)\n",
"\n",
"@pm.deterministic\n",
"def mu_imiquimod_112(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for imiquimod in RefID 112\n",
" return μ + θ[IMIQUIMOD] + ϵ[refid112]\n",
"\n",
"\n",
"y_imiquimod_112 = pm.Normal('y_imiquimod_112', mu_imiquimod_112, τ[IMIQUIMOD]**-1, \n",
" value=pm.logit(obs_imiquimod_112), observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Thayal 2012 (RefID 309)"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>44</th>\n",
" <td> 309</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> Complete involution (&gt; 90% response)</td>\n",
" <td> 2</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 90</td>\n",
" <td> 100</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>45</th>\n",
" <td> 309</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> reduction in size of 75-90 %</td>\n",
" <td> 4</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 75</td>\n",
" <td> 90</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>46</th>\n",
" <td> 309</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> reduction in size of 50 to 75 %</td>\n",
" <td> 3</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 75</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>47</th>\n",
" <td> 309</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> reduction in size of &lt;25 %</td>\n",
" <td> 1</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 25</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"44 309 propranolol 2 mg/kg oral 10 \n",
"45 309 propranolol 2 mg/kg oral 10 \n",
"46 309 propranolol 2 mg/kg oral 10 \n",
"47 309 propranolol 2 mg/kg oral 10 \n",
"\n",
" Outcome Outcome data/result Outcome data \\\n",
"44 Complete involution (> 90% response) 2 range \n",
"45 reduction in size of 75-90 % 4 range \n",
"46 reduction in size of 50 to 75 % 3 range \n",
"47 reduction in size of <25 % 1 range \n",
"\n",
" Point estimate sd Lower range Upper range Outcome data/result N \n",
"44 NaN NaN 90 100 2 \n",
"45 NaN NaN 75 90 4 \n",
"46 NaN NaN 50 75 3 \n",
"47 NaN NaN 0 25 1 "
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==309][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid309 = np.where(unique_studies==309)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_309(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for propanolol in RefID 309\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid309]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_309(μ=mu_propranolol_309, τ=τ):\n",
" # Response category probabilities for propanolol in RefID 309\n",
" c1 = norm.cdf(pm.logit(0.25), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[PROPRANOLOL]**-1)\n",
" c3 = norm.cdf(pm.logit(0.75), μ, τ[PROPRANOLOL]**-1)\n",
" c4 = norm.cdf(pm.logit(0.9), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, c3-c2, c4-c3, 1-c4 \n",
"\n",
"# Multinomial likelihood for propanolol in RefID 309\n",
"y_propanolol_309 = pm.Multinomial('y_propanolol_309', 10, p_propranolol_309, \n",
" value=[1, 0, 3, 4, 2], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Chambers 2012 (RefID 321)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>48</th>\n",
" <td> 321</td>\n",
" <td> timolol</td>\n",
" <td> 0.0025</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 13</td>\n",
" <td> reduction in size of &gt;50 %</td>\n",
" <td> 61.5</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 8</td>\n",
" </tr>\n",
" <tr>\n",
" <th>49</th>\n",
" <td> 321</td>\n",
" <td> timolol</td>\n",
" <td> 0.0025</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 13</td>\n",
" <td> reduction in size of 0-50 %</td>\n",
" <td> 30.8</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 50</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>50</th>\n",
" <td> 321</td>\n",
" <td> timolol</td>\n",
" <td> 0.0025</td>\n",
" <td> BID</td>\n",
" <td> topical</td>\n",
" <td> 13</td>\n",
" <td> enlarged in size</td>\n",
" <td> NaN</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>51</th>\n",
" <td> 321</td>\n",
" <td> control</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> reduction in size of &gt;50 %</td>\n",
" <td> 0</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 100</td>\n",
" <td> 0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>52</th>\n",
" <td> 321</td>\n",
" <td> control</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> reduction in size of 0-50 %</td>\n",
" <td> 10</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 50</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>53</th>\n",
" <td> 321</td>\n",
" <td> control</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 10</td>\n",
" <td> enlarged in size</td>\n",
" <td> NaN</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 9</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"48 321 timolol 0.0025 BID topical 13 \n",
"49 321 timolol 0.0025 BID topical 13 \n",
"50 321 timolol 0.0025 BID topical 13 \n",
"51 321 control NaN NaN NaN 10 \n",
"52 321 control NaN NaN NaN 10 \n",
"53 321 control NaN NaN NaN 10 \n",
"\n",
" Outcome Outcome data/result Outcome data \\\n",
"48 reduction in size of >50 % 61.5 range \n",
"49 reduction in size of 0-50 % 30.8 range \n",
"50 enlarged in size NaN range \n",
"51 reduction in size of >50 % 0 range \n",
"52 reduction in size of 0-50 % 10 range \n",
"53 enlarged in size NaN range \n",
"\n",
" Point estimate sd Lower range Upper range Outcome data/result N \n",
"48 NaN NaN 50 100 8 \n",
"49 NaN NaN 1 50 4 \n",
"50 NaN NaN 0 1 1 \n",
"51 NaN NaN 50 100 0 \n",
"52 NaN NaN 1 50 1 \n",
"53 NaN NaN 0 1 9 "
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==321][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid321 = np.where(unique_studies==321)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_timolol_321(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for timolol in RefID 321\n",
" return μ + θ[TIMOLOL] + ϵ[refid321]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_timolol_321(μ=mu_timolol_321, τ=τ):\n",
" # Response category probabilities for timolol in RefID 321\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[TIMOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[TIMOLOL]**-1)\n",
" return c1, c2-c1, 1 - c2\n",
"\n",
"# Binomial likelihood for propanolol in RefID 321\n",
"y_timolol_321 = pm.Multinomial('y_timolol_321', 13, p_timolol_321, value=[1, 4, 8], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_321(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for controls in RefID 321\n",
" return μ + ϵ[refid321]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_321(μ=mu_control_321, τ=τ):\n",
" # Response category probabilities for controls in RefID 321\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[-1]**-1)\n",
" c2 = norm.cdf(pm.logit(0.5), μ, τ[-1]**-1)\n",
" return c1, c2-c1, 1 - c2\n",
"\n",
"# Binomial likelihood for controls in RefID 321\n",
"y_control_321 = pm.Multinomial('y_control_321', 10, p_control_321, value=[9, 1, 0], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Bertrand 2011 (RefID 402)"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>54</th>\n",
" <td> 402</td>\n",
" <td> prednisone</td>\n",
" <td> 2.8</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 12</td>\n",
" <td> Visual Analog Scale - mean mm</td>\n",
" <td> 44.82</td>\n",
" <td> estimate</td>\n",
" <td> 44.82</td>\n",
" <td> 12.21</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>55</th>\n",
" <td> 402</td>\n",
" <td> propranolol</td>\n",
" <td> 2.7</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 12</td>\n",
" <td> Visual Analog Scale - mean mm</td>\n",
" <td> 78.73</td>\n",
" <td> estimate</td>\n",
" <td> 78.73</td>\n",
" <td> 22.47</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"54 402 prednisone 2.8 mg/kg oral 12 \n",
"55 402 propranolol 2.7 mg/kg oral 12 \n",
"\n",
" Outcome Outcome data/result Outcome data \\\n",
"54 Visual Analog Scale - mean mm 44.82 estimate \n",
"55 Visual Analog Scale - mean mm 78.73 estimate \n",
"\n",
" Point estimate sd Lower range Upper range Outcome data/result N \n",
"54 44.82 12.21 NaN NaN NaN \n",
"55 78.73 22.47 NaN NaN NaN "
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==402][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid402 = np.where(unique_studies==402)[0][0]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Find SD on the logit scale using the delta method"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"propanolol_402_var = (0.2247**2) / (0.7873 * (1-0.7873))**2\n",
"prednisone_402_var = (0.1221**2) / (0.4482 * (1-0.4482))**2"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_402(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 402\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid402]\n",
"\n",
"propranolol_obs_402 = pm.Normal('propranolol_obs_402', mu_propranolol_402, propanolol_402_var**-1, \n",
" value=pm.logit(0.7873), observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_prednisolone_402(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for prednisolone in RefID 402\n",
" return μ + θ[ORAL_STEROID] + ϵ[refid402]\n",
"\n",
"prednisolone_obs_402 = pm.Normal('prednisolone_obs_402', mu_prednisolone_402, prednisone_402_var**-1, \n",
" pm.logit(0.4482), observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Price 2011 (RefID 438)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>56</th>\n",
" <td> 438</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 68</td>\n",
" <td> Clearance ≥ 75%</td>\n",
" <td> 56</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 75</td>\n",
" <td> 100</td>\n",
" <td> 56</td>\n",
" </tr>\n",
" <tr>\n",
" <th>57</th>\n",
" <td> 438</td>\n",
" <td> propranolol</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 68</td>\n",
" <td> Clearance &lt;75%</td>\n",
" <td> 12</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 75</td>\n",
" <td> 12</td>\n",
" </tr>\n",
" <tr>\n",
" <th>58</th>\n",
" <td> 438</td>\n",
" <td> corticosteroids</td>\n",
" <td> 4</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 42</td>\n",
" <td> Clearance ≥ 75%</td>\n",
" <td> 12</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 75</td>\n",
" <td> 100</td>\n",
" <td> 12</td>\n",
" </tr>\n",
" <tr>\n",
" <th>59</th>\n",
" <td> 438</td>\n",
" <td> corticosteroids</td>\n",
" <td> 4</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 42</td>\n",
" <td> Clearance &lt;75%</td>\n",
" <td> 30</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 75</td>\n",
" <td> 30</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"56 438 propranolol 2 mg/kg oral 68 \n",
"57 438 propranolol 2 mg/kg oral 68 \n",
"58 438 corticosteroids 4 mg/kg oral 42 \n",
"59 438 corticosteroids 4 mg/kg oral 42 \n",
"\n",
" Outcome Outcome data/result Outcome data Point estimate sd \\\n",
"56 Clearance ≥ 75% 56 range NaN NaN \n",
"57 Clearance <75% 12 range NaN NaN \n",
"58 Clearance ≥ 75% 12 range NaN NaN \n",
"59 Clearance <75% 30 range NaN NaN \n",
"\n",
" Lower range Upper range Outcome data/result N \n",
"56 75 100 56 \n",
"57 0 75 12 \n",
"58 75 100 12 \n",
"59 0 75 30 "
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==438][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid438 = np.where(unique_studies==438)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propanolol_438(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for propanolol in RefID 438\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid438]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_438(μ=mu_propanolol_438, τ=τ):\n",
" # Response category probabilities for propanolol in RefID 438\n",
" return 1 - norm.cdf(pm.logit(0.75), μ, τ[PROPRANOLOL]**-1)\n",
"\n",
"# Binomial likelihood for propanolol in RefID 5\n",
"y_propranolol_438 = pm.Binomial('y_propranolol_438', 68, p_propranolol_438, \n",
" value=56, observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_corticosteriod_438(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for corticosteriods in RefID 438\n",
" return μ + θ[ORAL_STEROID] + ϵ[refid438]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_corticosteriod_438(μ=mu_corticosteriod_438, τ=τ):\n",
" # Response category probabilities for corticosteriods in RefID 438\n",
" return 1 - norm.cdf(pm.logit(0.75), μ, τ[ORAL_STEROID]**-1)\n",
"\n",
"# Binomial likelihood for propanolol in RefID 5\n",
"y_corticosteriod_438 = pm.Binomial('y_corticosteriod_438', 42, p_corticosteriod_438, \n",
" value=12, observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Yu 2013 (RefID 3451)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>60</th>\n",
" <td> 3451</td>\n",
" <td> timolol</td>\n",
" <td> NR</td>\n",
" <td> NR</td>\n",
" <td> topical</td>\n",
" <td> 101</td>\n",
" <td> Response - ineffective</td>\n",
" <td> 7.9</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 8</td>\n",
" </tr>\n",
" <tr>\n",
" <th>61</th>\n",
" <td> 3451</td>\n",
" <td> timolol</td>\n",
" <td> NR</td>\n",
" <td> NR</td>\n",
" <td> topical</td>\n",
" <td> 101</td>\n",
" <td> Response - controlled growth</td>\n",
" <td> 35.6</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 36</td>\n",
" </tr>\n",
" <tr>\n",
" <th>62</th>\n",
" <td> 3451</td>\n",
" <td> timolol</td>\n",
" <td> NR</td>\n",
" <td> NR</td>\n",
" <td> topical</td>\n",
" <td> 101</td>\n",
" <td> Response - promoted regression</td>\n",
" <td> 56.4</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 99</td>\n",
" <td> 57</td>\n",
" </tr>\n",
" <tr>\n",
" <th>63</th>\n",
" <td> 3451</td>\n",
" <td> timolol</td>\n",
" <td> NR</td>\n",
" <td> NR</td>\n",
" <td> topical</td>\n",
" <td> 101</td>\n",
" <td> Complete regression</td>\n",
" <td> 12</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 12</td>\n",
" </tr>\n",
" <tr>\n",
" <th>64</th>\n",
" <td> 3451</td>\n",
" <td> No treatment - observation</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 23</td>\n",
" <td> Response - ineffective</td>\n",
" <td> 65.2</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 15</td>\n",
" </tr>\n",
" <tr>\n",
" <th>65</th>\n",
" <td> 3451</td>\n",
" <td> No treatment - observation</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 23</td>\n",
" <td> Response - controlled growth</td>\n",
" <td> 30.4</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 1</td>\n",
" <td> 7</td>\n",
" </tr>\n",
" <tr>\n",
" <th>66</th>\n",
" <td> 3451</td>\n",
" <td> No treatment - observation</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" <td> 23</td>\n",
" <td> Response - promotedregression</td>\n",
" <td> 4.3</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 1</td>\n",
" <td> 99</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form \\\n",
"60 3451 timolol NR NR topical \n",
"61 3451 timolol NR NR topical \n",
"62 3451 timolol NR NR topical \n",
"63 3451 timolol NR NR topical \n",
"64 3451 No treatment - observation NaN NaN NaN \n",
"65 3451 No treatment - observation NaN NaN NaN \n",
"66 3451 No treatment - observation NaN NaN NaN \n",
"\n",
" N at follow-up Outcome Outcome data/result \\\n",
"60 101 Response - ineffective 7.9 \n",
"61 101 Response - controlled growth 35.6 \n",
"62 101 Response - promoted regression 56.4 \n",
"63 101 Complete regression 12 \n",
"64 23 Response - ineffective 65.2 \n",
"65 23 Response - controlled growth 30.4 \n",
"66 23 Response - promotedregression 4.3 \n",
"\n",
" Outcome data Point estimate sd Lower range Upper range \\\n",
"60 range NaN NaN 0 1 \n",
"61 range NaN NaN 0 1 \n",
"62 range NaN NaN 1 99 \n",
"63 range NaN NaN 99 100 \n",
"64 range NaN NaN 0 1 \n",
"65 range NaN NaN 0 1 \n",
"66 range NaN NaN 1 99 \n",
"\n",
" Outcome data/result N \n",
"60 8 \n",
"61 36 \n",
"62 57 \n",
"63 12 \n",
"64 15 \n",
"65 7 \n",
"66 1 "
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==3451][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid3451 = np.where(unique_studies==3451)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_timolol_3451(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for timolol in RefID 3451\n",
" return μ + θ[TIMOLOL] + ϵ[refid3451]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_timolol_3451(μ=mu_timolol_3451, τ=τ):\n",
" # Response category probabilities for timolol in RefID 3451\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[TIMOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.99), μ, τ[TIMOLOL]**-1)\n",
" return c1, c2-c1, 1 - c2\n",
"\n",
"# Binomial likelihood for propanolol in RefID 3451\n",
"y_timolol_3451 = pm.Multinomial('y_timolol_3451', 101, p_timolol_3451, value=[44, 57-12, 12], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_control_3451(μ=μ, ϵ=ϵ):\n",
" # Mean response on the logit scale for controls in RefID 3451\n",
" return μ + ϵ[refid3451]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_control_3451(μ=mu_control_3451, τ=τ):\n",
" # Response category probabilities for timolol in RefID 3451\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[-1]**-1)\n",
" c2 = norm.cdf(pm.logit(0.99), μ, τ[-1]**-1)\n",
" return c1, c2-c1, 1 - c2\n",
"\n",
"# Binomial likelihood for propanolol in RefID 3451\n",
"y_control_3451 = pm.Multinomial('y_control_3451', 23, p_control_3451, value=[22, 1, 0], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Awadein 2011 (RefID 3522)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>67</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Size of hemangioma (cm2)</td>\n",
" <td> 3.6 ± 2.6</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>68</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Response rate - Complete resolution achieved</td>\n",
" <td> 42</td>\n",
" <td> scalar</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>69</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Response rate - Sustained plateau, with &gt;50% r...</td>\n",
" <td> 25</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 99</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>70</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Response rate - &lt;50% reduction in size of hema...</td>\n",
" <td> 17</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 50</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>71</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Resistant to treatment</td>\n",
" <td> 17</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>72</th>\n",
" <td> 3522</td>\n",
" <td> propranolol</td>\n",
" <td> 1</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 12</td>\n",
" <td> Rebound growth</td>\n",
" <td> 4</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>73</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Size of hemangioma (cm2)</td>\n",
" <td> 3.7 ± 2.5</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>74</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Response rate - Complete resolution achieved</td>\n",
" <td> 40</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 99</td>\n",
" <td> 100</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>75</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Response rate - Sustained plateau, with &gt;50% r...</td>\n",
" <td> 20</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 50</td>\n",
" <td> 99</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>76</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Response rate - &lt;50% reduction in size of hema...</td>\n",
" <td> 20</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 50</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>77</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Resistant to treatment</td>\n",
" <td> 20</td>\n",
" <td> range</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 0</td>\n",
" <td> 0</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>78</th>\n",
" <td> 3522</td>\n",
" <td> triamcinolone</td>\n",
" <td> 40</td>\n",
" <td> mg/ml</td>\n",
" <td> intralesional</td>\n",
" <td> 10</td>\n",
" <td> Rebound growth</td>\n",
" <td> 3</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"67 3522 propranolol 1 mg/ml intralesional 12 \n",
"68 3522 propranolol 1 mg/ml intralesional 12 \n",
"69 3522 propranolol 1 mg/ml intralesional 12 \n",
"70 3522 propranolol 1 mg/ml intralesional 12 \n",
"71 3522 propranolol 1 mg/ml intralesional 12 \n",
"72 3522 propranolol 1 mg/ml intralesional 12 \n",
"73 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"74 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"75 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"76 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"77 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"78 3522 triamcinolone 40 mg/ml intralesional 10 \n",
"\n",
" Outcome Outcome data/result \\\n",
"67 Size of hemangioma (cm2) 3.6 ± 2.6 \n",
"68 Response rate - Complete resolution achieved 42 \n",
"69 Response rate - Sustained plateau, with >50% r... 25 \n",
"70 Response rate - <50% reduction in size of hema... 17 \n",
"71 Resistant to treatment 17 \n",
"72 Rebound growth 4 \n",
"73 Size of hemangioma (cm2) 3.7 ± 2.5 \n",
"74 Response rate - Complete resolution achieved 40 \n",
"75 Response rate - Sustained plateau, with >50% r... 20 \n",
"76 Response rate - <50% reduction in size of hema... 20 \n",
"77 Resistant to treatment 20 \n",
"78 Rebound growth 3 \n",
"\n",
" Outcome data Point estimate sd Lower range Upper range \\\n",
"67 NaN NaN NaN NaN NaN \n",
"68 scalar NaN NaN 99 100 \n",
"69 range NaN NaN 50 99 \n",
"70 range NaN NaN 0 50 \n",
"71 NaN NaN NaN NaN NaN \n",
"72 NaN NaN NaN NaN NaN \n",
"73 NaN NaN NaN NaN NaN \n",
"74 range NaN NaN 99 100 \n",
"75 range NaN NaN 50 99 \n",
"76 range NaN NaN 0 50 \n",
"77 range NaN NaN 0 0 \n",
"78 NaN NaN NaN NaN NaN \n",
"\n",
" Outcome data/result N \n",
"67 NaN \n",
"68 5 \n",
"69 3 \n",
"70 2 \n",
"71 2 \n",
"72 4 \n",
"73 NaN \n",
"74 4 \n",
"75 2 \n",
"76 2 \n",
"77 2 \n",
"78 3 "
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==3522][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid3522 = np.where(unique_studies==3522)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_intralesional_3522(μ=μ, θ=θ, ϕ=ϕ, ϵ=ϵ):\n",
" # Mean response on the logit scale for intralesional propanolol in RefID 3522\n",
" return μ + θ[PROPRANOLOL] + ϕ + ϵ[refid3522]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_propranolol_intralesional_3522(μ=mu_propranolol_intralesional_3522, τ=τ):\n",
" # Response category probabilities for intralesional propanolol in RefID 3522\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[PROPRANOLOL]**-1)\n",
" c2 = norm.cdf(pm.logit(0.50), μ, τ[PROPRANOLOL]**-1)\n",
" c3 = norm.cdf(pm.logit(0.99), μ, τ[PROPRANOLOL]**-1)\n",
" return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# Multinomial likelihood for intralesional propanolol in RefID 3522\n",
"y_propanolol_intralesional_3522 = pm.Multinomial('y_propanolol_intralesional_3522', 12, p_propranolol_intralesional_43, \n",
" value=[2, 2, 3, 5], observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_triamcinolone_3522(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for triamcinolone in RefID 3522\n",
" return μ + θ[TRIAMCILONE] + ϵ[refid3522]\n",
"\n",
"\n",
"@pm.deterministic\n",
"def p_triamcinolone_3522(μ=mu_triamcinolone_3522, τ=τ):\n",
" # Response category probabilities for triamcinolone in RefID 3522\n",
" c1 = norm.cdf(pm.logit(0.01), μ, τ[TRIAMCILONE]**-1)\n",
" c2 = norm.cdf(pm.logit(0.50), μ, τ[TRIAMCILONE]**-1)\n",
" c3 = norm.cdf(pm.logit(0.99), μ, τ[TRIAMCILONE]**-1)\n",
" return c1, c2-c1, c3-c2, 1-c3 \n",
"\n",
"# Multinomial likelihood for triamcinolone in RefID 3522\n",
"y_triamcinolone_3522 = pm.Multinomial('y_triamcinolone_3522', 10, p_triamcinolone_3522, \n",
" value=[2, 2, 2, 4], observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Pope 2012 (RefID 3723)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>79</th>\n",
" <td> 3723</td>\n",
" <td> nadolol</td>\n",
" <td> up to 4.0</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> percentage IH shrinkage</td>\n",
" <td> 97 ± 3.05</td>\n",
" <td> estimate</td>\n",
" <td> 97</td>\n",
" <td> 3.05</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>80</th>\n",
" <td> 3723</td>\n",
" <td> propranolol</td>\n",
" <td> 2 to 3</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 9</td>\n",
" <td> percentage IH shrinkage</td>\n",
" <td> 86 ± 14.82</td>\n",
" <td> estimate</td>\n",
" <td> 86</td>\n",
" <td> 14.82</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form N at follow-up \\\n",
"79 3723 nadolol up to 4.0 mg/kg oral 10 \n",
"80 3723 propranolol 2 to 3 mg/kg oral 9 \n",
"\n",
" Outcome Outcome data/result Outcome data Point estimate \\\n",
"79 percentage IH shrinkage 97 ± 3.05 estimate 97 \n",
"80 percentage IH shrinkage 86 ± 14.82 estimate 86 \n",
"\n",
" sd Lower range Upper range Outcome data/result N \n",
"79 3.05 NaN NaN NaN \n",
"80 14.82 NaN NaN NaN "
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==3723][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid3723 = np.where(unique_studies==3723)[0][0]"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"propanolol_3723_var = (0.1482**2) / (0.86 * (1-0.86))**2\n",
"nadolol_3723_var = (0.0305**2) / (0.97 * (1-0.97))**2"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_propranolol_3723(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 402\n",
" return μ + θ[PROPRANOLOL] + ϵ[refid3723]\n",
"\n",
"propranolol_obs_3723 = pm.Normal('propranolol_obs_3723', mu_propranolol_3723, propanolol_3723_var**-1, \n",
" value=pm.logit(0.86), observed=True)"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_nadolol_3723(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 402\n",
" return μ + θ[NADOLOL] + ϵ[refid3723]\n",
"\n",
"nadolol_obs_3723 = pm.Normal('nadolol_obs_3723', mu_nadolol_3723, nadolol_3723_var**-1, \n",
" value=pm.logit(0.97), observed=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Pope 2007 (RefID 1180)"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>REFID</th>\n",
" <th>Arm</th>\n",
" <th>Dosage</th>\n",
" <th>Dosage amount</th>\n",
" <th>Dosage form</th>\n",
" <th>N at follow-up</th>\n",
" <th>Outcome</th>\n",
" <th>Outcome data/result</th>\n",
" <th>Outcome data</th>\n",
" <th>Point estimate</th>\n",
" <th>sd</th>\n",
" <th>Lower range</th>\n",
" <th>Upper range</th>\n",
" <th>Outcome data/result N</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>81</th>\n",
" <td> 1180</td>\n",
" <td> Methylprednisolone</td>\n",
" <td> 30</td>\n",
" <td> mg/kg</td>\n",
" <td> intravenous</td>\n",
" <td> 10</td>\n",
" <td> Change in size measured using visual acuity sc...</td>\n",
" <td> -1.5 (-35 to 22)</td>\n",
" <td> difference</td>\n",
" <td> -1.5</td>\n",
" <td> NaN</td>\n",
" <td>-35</td>\n",
" <td> 22</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>82</th>\n",
" <td> 1180</td>\n",
" <td> prednisolone</td>\n",
" <td> 2</td>\n",
" <td> mg/kg</td>\n",
" <td> oral</td>\n",
" <td> 10</td>\n",
" <td> Change in size measured using VAS at AGE 1 year</td>\n",
" <td> 50 (35 to 67)</td>\n",
" <td> difference</td>\n",
" <td> 50.0</td>\n",
" <td> NaN</td>\n",
" <td> 35</td>\n",
" <td> 67</td>\n",
" <td> NaN</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" REFID Arm Dosage Dosage amount Dosage form \\\n",
"81 1180 Methylprednisolone 30 mg/kg intravenous \n",
"82 1180 prednisolone 2 mg/kg oral \n",
"\n",
" N at follow-up Outcome \\\n",
"81 10 Change in size measured using visual acuity sc... \n",
"82 10 Change in size measured using VAS at AGE 1 year \n",
"\n",
" Outcome data/result Outcome data Point estimate sd Lower range \\\n",
"81 -1.5 (-35 to 22) difference -1.5 NaN -35 \n",
"82 50 (35 to 67) difference 50.0 NaN 35 \n",
"\n",
" Upper range Outcome data/result N \n",
"81 22 NaN \n",
"82 67 NaN "
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kq2_outcomes[kq2_outcomes.REFID==1180][subset_cols]"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"refid1180 = np.where(unique_studies==1180)[0][0]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Assuming the width of the interquartile range is 1.35 SD."
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"prednisolone_sd_1180 = (0.67 - 0.35) / 1.35\n",
"methylprednisolone_sd_1180 = (0.22 + 0.35) / 1.35"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"methylprednisolone_1180_latent = pm.Normal('methylprednisolone_1180_latent', -0.015, \n",
" methylprednisolone_sd_1180**-2, size=10)"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"methylprednisolone_1180_trunc = pm.Lambda('methylprednisolone_1180_trunc', \n",
" lambda y=methylprednisolone_1180_latent: np.clip(y, 1e-6, 1-1e-6))"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"prednisolone_1180_latent = pm.Normal('prednisolone_1180_latent', 0.5, \n",
" prednisolone_sd_1180**-2, size=10)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"prednisolone_1180_trunc = pm.Lambda('prednisolone_1180_trunc', \n",
" lambda y=prednisolone_1180_latent: np.clip(y, 1e-6, 1-1e-6)).value"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_prednisolone_1180(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for prednisolone in RefID 1180\n",
" return μ + θ[ORAL_STEROID] + ϵ[refid1180]\n",
"\n",
"@pm.potential\n",
"def prednisolone_obs_1180(μ=mu_prednisolone_1180, τ=τ, y=prednisolone_1180_trunc):\n",
" return pm.normal_like(pm.logit(y), μ, τ[ORAL_STEROID])"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def mu_methylprednisolone_1180(μ=μ, θ=θ, ϵ=ϵ):\n",
" # Mean response on the logit scale for oral propanolol in RefID 1180\n",
" return μ + θ[METHYLPREDNISOLONE] + ϵ[refid1180]\n",
"\n",
"@pm.potential\n",
"def methylprednisolone_obs_1180(μ=mu_methylprednisolone_1180, τ=τ, y=methylprednisolone_1180_trunc):\n",
" return pm.normal_like(pm.logit(y), μ, τ[METHYLPREDNISOLONE])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Expected clearance rates for each treatment"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def expected_clearance(μ=μ, θ=θ, ϕ=ϕ):\n",
" p = np.append(pm.invlogit(μ), pm.invlogit(μ + θ[0] + ϕ))\n",
" return np.append(p, pm.invlogit(μ + θ))"
]
},
{
"cell_type": "code",
"execution_count": 101,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"@pm.deterministic\n",
"def predicted_clearance(μ=μ, θ=θ, ϕ=ϕ, τ=τ):\n",
" m = np.append(np.append(μ, μ + θ[0] + ϕ), μ + θ)\n",
" T = np.append(np.append(τ[-1], τ[0]), τ[:-1]) \n",
" y = pm.rnormal(m, T)\n",
" return pm.invlogit(y)"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"best = pm.Lambda('best', lambda b=expected_clearance: (b==b.max()).astype(int))"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"rate_labels = ['Control', \n",
" 'Intralesional propanolol', \n",
" 'Oral propanolol', \n",
" 'Atenolol', \n",
" 'Timolol', \n",
" 'Triamcilone', \n",
" 'Oral steroid',\n",
" 'Imiquimod', \n",
" 'Nadolol', \n",
" 'Methylprednisolone']"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" [- 2% ] 5282 of 200000 complete in 104.1 sec"
]
}
],
"source": [
"iterations, burn = 200000, 190000\n",
"\n",
"M = pm.MCMC(locals())\n",
"M.use_step_method(pm.AdaptiveMetropolis, σ)\n",
"M.sample(iterations, burn)"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"M.sample(iterations, burn)"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"best:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t0.001 0.037 0.001 [ 0. 0.]\n",
"\t0.134 0.341 0.028 [ 0. 1.]\n",
"\t0.137 0.344 0.03 [ 0. 1.]\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t0.727 0.445 0.039 [ 0. 1.]\n",
"\t0.0 0.0 0.0 [ 0. 0.]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 0.0 0.0 1.0\n",
"\t0.0 0.0 0.0 0.0 1.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t0.0 0.0 1.0 1.0 1.0\n",
"\t0.0 0.0 0.0 0.0 0.0\n",
"\t\n"
]
}
],
"source": [
"best.summary()"
]
},
{
"cell_type": "code",
"execution_count": 125,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Plotting θ_0\n",
"Plotting θ_1\n",
"Plotting θ_2\n",
"Plotting θ_3\n",
"Plotting θ_4\n",
"Plotting θ_5\n",
"Plotting θ_6\n",
"Plotting θ_7\n"
]
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x1174b89e8>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x117fd3d68>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pm.Matplot.plot(θ)"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Plotting σ_0\n",
"Plotting σ_1\n",
"Plotting σ_2\n",
"Plotting σ_3\n",
"Plotting σ_4\n",
"Plotting σ_5\n",
"Plotting σ_6\n",
"Plotting σ_7\n",
"Plotting σ_8\n"
]
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x11829bdd8>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x1170507f0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x117b77a58>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pm.Matplot.plot(σ)"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"μ:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t-2.022 0.682 0.067 [-3.296 -1.043]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t-3.321 -2.571 -1.825 -1.478 -1.048\n",
"\t\n"
]
}
],
"source": [
"μ.summary()"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"θ:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t5.31 0.772 0.071 [ 3.938 6.841]\n",
"\t5.069 1.371 0.133 [ 2.412 7.345]\n",
"\t3.778 0.486 0.045 [ 2.798 4.732]\n",
"\t3.154 0.441 0.037 [ 2.326 3.942]\n",
"\t0.847 0.892 0.087 [-0.587 2.402]\n",
"\t2.679 0.608 0.056 [ 1.503 3.942]\n",
"\t6.607 1.275 0.124 [ 4.273 8.866]\n",
"\t1.195 0.976 0.096 [-0.144 3.149]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t3.821 4.795 5.26 5.875 6.763\n",
"\t2.075 4.049 5.261 6.226 7.076\n",
"\t2.852 3.444 3.779 4.115 4.82\n",
"\t2.376 2.844 3.133 3.449 4.078\n",
"\t-0.696 0.087 0.818 1.622 2.358\n",
"\t1.457 2.261 2.718 3.058 3.916\n",
"\t4.38 5.689 6.525 7.469 9.13\n",
"\t-0.135 0.359 1.073 1.881 3.264\n",
"\t\n"
]
}
],
"source": [
"θ.summary()"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(array([ 53., 199., 607., 1059., 1206., 1898., 1904., 1560.,\n",
" 1141., 373.]),\n",
" array([ 0.2064767 , 0.28074814, 0.35501958, 0.42929102, 0.50356245,\n",
" 0.57783389, 0.65210533, 0.72637677, 0.80064821, 0.87491964,\n",
" 0.94919108]),\n",
" <a list of 10 Patch objects>)"
]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"/vbnbs3MK/o/Ze9msL51wC2ZeWJTM87FDNb3WuA1tF479D/Axe2jKCXMYH3vpPWitr3ABzPzK81M\n",
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x11721cb38>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pl.hist(pm.invlogit(μ.trace() + θ.trace()[:, 5]))"
]
},
{
"cell_type": "code",
"execution_count": 136,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Could not calculate Gelman-Rubin statistics. Requires multiple chains of equal length.\n"
]
},
{
"data": {
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"YII=\n"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x1174fa7f0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sigma_labels = ['Propanolol', \n",
" 'Atenolol', \n",
" 'Timolol', \n",
" 'Triamcilone', \n",
" 'Oral steroid',\n",
" 'Imiquimod', \n",
" 'Nadolol', \n",
" 'Methylprednisolone',\n",
" 'Control']\n",
"\n",
"pm.Matplot.summary_plot(σ, custom_labels=sigma_labels, \n",
" vline_pos=-1, main='Standard deviation of effects')"
]
},
{
"cell_type": "code",
"execution_count": 138,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Could not calculate Gelman-Rubin statistics. Requires multiple chains of equal length.\n"
]
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x117906dd8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pm.Matplot.summary_plot(θ, custom_labels=sigma_labels, \n",
" vline_pos=-10, main='Expected effect size (relative to control)')"
]
},
{
"cell_type": "code",
"execution_count": 121,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Could not calculate Gelman-Rubin statistics. Requires multiple chains of equal length.\n"
]
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x118374b00>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pm.Matplot.summary_plot(predicted_clearance, custom_labels=rate_labels, vline_pos=-1)"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Could not calculate Gelman-Rubin statistics. Requires multiple chains of equal length.\n"
]
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x117214240>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pm.Matplot.summary_plot(expected_clearance, custom_labels=rate_labels, \n",
" main='Expected clearance (%)', vline_pos=-1)"
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"μ:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t-2.022 0.682 0.067 [-3.296 -1.043]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t-3.321 -2.571 -1.825 -1.478 -1.048\n",
"\t\n"
]
}
],
"source": [
"μ.summary()"
]
},
{
"cell_type": "code",
"execution_count": 124,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"σ:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t2.651 0.182 0.014 [ 2.282 3.007]\n",
"\t1.842 0.605 0.05 [ 0.431 2.776]\n",
"\t1.503 0.033 0.003 [ 1.442 1.574]\n",
"\t2.364 0.314 0.026 [ 1.852 2.951]\n",
"\t1.363 0.15 0.012 [ 1.102 1.687]\n",
"\t0.757 0.131 0.011 [ 0.515 0.991]\n",
"\t1.269 0.989 0.079 [ 0.01 3.137]\n",
"\t0.375 0.339 0.032 [ 0.04 1.057]\n",
"\t1.106 0.122 0.011 [ 0.885 1.316]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t2.32 2.542 2.635 2.767 3.056\n",
"\t0.536 1.422 1.909 2.212 2.944\n",
"\t1.443 1.482 1.501 1.525 1.575\n",
"\t1.908 2.148 2.31 2.56 3.035\n",
"\t1.129 1.251 1.344 1.45 1.724\n",
"\t0.523 0.664 0.747 0.859 1.018\n",
"\t0.061 0.442 1.076 1.875 3.683\n",
"\t0.042 0.106 0.272 0.52 1.248\n",
"\t0.892 1.012 1.097 1.186 1.349\n",
"\t\n"
]
}
],
"source": [
"σ.summary()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.4.2"
}
},
"nbformat": 4,
"nbformat_minor": 0
}
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