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josef-pkt/example_tukeyhsd.py

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Using Tukey HSD in statsmodels
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example_tukeyhsd.py
{
"metadata": {
"name": "example_tukeyhsd"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Multiple Comparison in OneWay ANOVA"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from statsmodels.compatnp.py3k import BytesIO, asbytes # for python 3 compatibility\n",
"import numpy as np\n",
"import statsmodels.stats.multicomp as multi"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 14
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following we have two datasets for which we would like to test whether the response differs across different levels of the explanatory variable."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ss = '''\\\n",
" 43.9 1 1\n",
" 39.0 1 2\n",
" 46.7 1 3\n",
" 43.8 1 4\n",
" 44.2 1 5\n",
" 47.7 1 6\n",
" 43.6 1 7\n",
" 38.9 1 8\n",
" 43.6 1 9\n",
" 40.0 1 10\n",
" 89.8 2 1\n",
" 87.1 2 2\n",
" 92.7 2 3\n",
" 90.6 2 4\n",
" 87.7 2 5\n",
" 92.4 2 6\n",
" 86.1 2 7\n",
" 88.1 2 8\n",
" 90.8 2 9\n",
" 89.1 2 10\n",
" 68.4 3 1\n",
" 69.3 3 2\n",
" 68.5 3 3\n",
" 66.4 3 4\n",
" 70.0 3 5\n",
" 68.1 3 6\n",
" 70.6 3 7\n",
" 65.2 3 8\n",
" 63.8 3 9\n",
" 69.2 3 10\n",
" 36.2 4 1\n",
" 45.2 4 2\n",
" 40.7 4 3\n",
" 40.5 4 4\n",
" 39.3 4 5\n",
" 40.3 4 6\n",
" 43.2 4 7\n",
" 38.7 4 8\n",
" 40.9 4 9\n",
" 39.7 4 10'''\n",
"\n",
"#idx Treatment StressReduction\n",
"ss2 = '''\\\n",
"1 mental 2\n",
"2 mental 2\n",
"3 mental 3\n",
"4 mental 4\n",
"5 mental 4\n",
"6 mental 5\n",
"7 mental 3\n",
"8 mental 4\n",
"9 mental 4\n",
"10 mental 4\n",
"11 physical 4\n",
"12 physical 4\n",
"13 physical 3\n",
"14 physical 5\n",
"15 physical 4\n",
"16 physical 1\n",
"17 physical 1\n",
"18 physical 2\n",
"19 physical 3\n",
"20 physical 3\n",
"21 medical 1\n",
"22 medical 2\n",
"23 medical 2\n",
"24 medical 2\n",
"25 medical 3\n",
"26 medical 2\n",
"27 medical 3\n",
"28 medical 1\n",
"29 medical 3\n",
"30 medical 1'''\n",
"\n",
"#accommodate recfromtxt for python 3.2, requires bytes\n",
"ss = asbytes(ss)\n",
"ss2 = asbytes(ss2)\n",
"\n",
"\n",
"dta = np.recfromtxt(BytesIO(ss), names=(\"Rust\",\"Brand\",\"Replication\"))\n",
"dta2 = np.recfromtxt(BytesIO(ss2), names = (\"idx\", \"Treatment\", \"StressReduction\"))\n",
"import pandas as pd\n",
"dta1 = pd.read_csv(BytesIO(ss), delimiter=' +', names=(\"Rust\",\"Brand\",\"Replication\"))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 15
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The fo"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dta1.head()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"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>Rust</th>\n",
" <th>Brand</th>\n",
" <th>Replication</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td> 43.9</td>\n",
" <td> 1</td>\n",
" <td> 1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td> 39.0</td>\n",
" <td> 1</td>\n",
" <td> 2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td> 46.7</td>\n",
" <td> 1</td>\n",
" <td> 3</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td> 43.8</td>\n",
" <td> 1</td>\n",
" <td> 4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td> 44.2</td>\n",
" <td> 1</td>\n",
" <td> 5</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"output_type": "pyout",
"prompt_number": 16,
"text": [
" Rust Brand Replication\n",
"0 43.9 1 1\n",
"1 39.0 1 2\n",
"2 46.7 1 3\n",
"3 43.8 1 4\n",
"4 44.2 1 5"
]
}
],
"prompt_number": 16
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#help(multi.MultiComparison)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 38
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#What's the nicest, fastest way to get F-test for oneway anova?\n",
"\n",
"The F-test in the following regression shows that the null hypothesis that all coefficients are zero, is strongly rejected with a p-value of 1e-33. This means in terms of a one way anova, that we can reject the joint hypothesis that all means of the response are the same across each explanatory variable, in this case the brand.\n",
"\n",
"In the following we use Tukey HSD to test each pairwise comparison. We reject that all means are the same, but we would like to know if some pairs of brands have the same mean."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import statsmodels.formula.api as smf\n",
"res = smf.ols(\"Rust ~ C(Brand, Treatment(1))\", dta1).fit()\n",
"print res.summary()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: Rust R-squared: 0.986\n",
"Model: OLS Adj. R-squared: 0.985\n",
"Method: Least Squares F-statistic: 866.1\n",
"Date: Sat, 05 Apr 2014 Prob (F-statistic): 1.34e-33\n",
"Time: 21:32:38 Log-Likelihood: -90.946\n",
"No. Observations: 40 AIC: 189.9\n",
"Df Residuals: 36 BIC: 196.6\n",
"Df Model: 3 \n",
"Covariance Type: nonrobust \n",
"===============================================================================================\n",
" coef std err t P>|t| [95.0% Conf. Int.]\n",
"-----------------------------------------------------------------------------------------------\n",
"Intercept 43.1400 0.784 55.056 0.000 41.551 44.729\n",
"C(Brand, Treatment(1))[T.2] 46.3000 1.108 41.782 0.000 44.053 48.547\n",
"C(Brand, Treatment(1))[T.3] 24.8100 1.108 22.389 0.000 22.563 27.057\n",
"C(Brand, Treatment(1))[T.4] -2.6700 1.108 -2.409 0.021 -4.917 -0.423\n",
"==============================================================================\n",
"Omnibus: 0.395 Durbin-Watson: 2.608\n",
"Prob(Omnibus): 0.821 Jarque-Bera (JB): 0.555\n",
"Skew: -0.101 Prob(JB): 0.758\n",
"Kurtosis: 2.459 Cond. No. 4.79\n",
"==============================================================================\n",
"\n",
"Warnings:\n",
"[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
]
}
],
"prompt_number": 39
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mc = multi.MultiComparison(dta['Rust'], dta['Brand'])\n",
"res = mc.tukeyhsd()\n",
"print(res.summary())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Multiple Comparison of Means - Tukey HSD,FWER=0.05\n",
"===============================================\n",
"group1 group2 meandiff lower upper reject\n",
"-----------------------------------------------\n",
" 0 1 46.3 43.3155 49.2845 True \n",
" 0 2 24.81 21.8255 27.7945 True \n",
" 0 3 -2.67 -5.6545 0.3145 False \n",
" 1 2 -21.49 -24.4745 -18.5055 True \n",
" 1 3 -48.97 -51.9545 -45.9855 True \n",
" 2 3 -27.48 -30.4645 -24.4955 True \n",
"-----------------------------------------------\n"
]
}
],
"prompt_number": 18
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mc = multi.MultiComparison(dta1['Rust'], dta1['Brand'])#.values)\n",
"res = mc.tukeyhsd()\n",
"print(res.summary())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Multiple Comparison of Means - Tukey HSD,FWER=0.05\n",
"===============================================\n",
"group1 group2 meandiff lower upper reject\n",
"-----------------------------------------------\n",
" 0 1 46.3 43.3155 49.2845 True \n",
" 0 2 24.81 21.8255 27.7945 True \n",
" 0 3 -2.67 -5.6545 0.3145 False \n",
" 1 2 -21.49 -24.4745 -18.5055 True \n",
" 1 3 -48.97 -51.9545 -45.9855 True \n",
" 2 3 -27.48 -30.4645 -24.4955 True \n",
"-----------------------------------------------\n"
]
}
],
"prompt_number": 19
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#vars(res)\n",
"res.groupsunique\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "pyout",
"prompt_number": 20,
"text": [
"Brand\n",
"0 1\n",
"18 2\n",
"29 3\n",
"38 4\n",
"Name: Brand, dtype: int64"
]
}
],
"prompt_number": 20
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can perform the same hypothesis tests for the second data set."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Bug: this doesn't work with recarray, why?\n",
"#res = smf.ols(\"StressReduction ~ C(Treatment, Treatment(1))\", \n",
"# data={'StressReduction':dta2['StressReduction'], 'Treatment':dta2['Treatment']}).fit()\n",
"res = smf.ols(\"StressReduction ~ Treatment\", dta2).fit()\n",
"print res.summary()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: StressReduction R-squared: 0.277\n",
"Model: OLS Adj. R-squared: 0.223\n",
"Method: Least Squares F-statistic: 5.164\n",
"Date: Sat, 05 Apr 2014 Prob (F-statistic): 0.0126\n",
"Time: 21:52:28 Log-Likelihood: -42.816\n",
"No. Observations: 30 AIC: 91.63\n",
"Df Residuals: 27 BIC: 95.84\n",
"Df Model: 2 \n",
"Covariance Type: nonrobust \n",
"=========================================================================================\n",
" coef std err t P>|t| [95.0% Conf. Int.]\n",
"-----------------------------------------------------------------------------------------\n",
"Intercept 2.0000 0.336 5.951 0.000 1.310 2.690\n",
"Treatment[T.mental] 1.5000 0.475 3.156 0.004 0.525 2.475\n",
"Treatment[T.physical] 1.0000 0.475 2.104 0.045 0.025 1.975\n",
"==============================================================================\n",
"Omnibus: 0.893 Durbin-Watson: 1.385\n",
"Prob(Omnibus): 0.640 Jarque-Bera (JB): 0.918\n",
"Skew: -0.293 Prob(JB): 0.632\n",
"Kurtosis: 2.374 Cond. No. 3.73\n",
"==============================================================================\n",
"\n",
"Warnings:\n",
"[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
]
}
],
"prompt_number": 45
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mc2 = multi.MultiComparison(dta2['StressReduction'], dta2['Treatment'])\n",
"res2 = mc2.tukeyhsd()\n",
"print(res2.summary())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Multiple Comparison of Means - Tukey HSD,FWER=0.05\n",
"============================================\n",
"group1 group2 meandiff lower upper reject\n",
"--------------------------------------------\n",
" 0 1 1.5 0.3217 2.6783 True \n",
" 0 2 1.0 -0.1783 2.1783 False \n",
" 1 2 -0.5 -1.6783 0.6783 False \n",
"--------------------------------------------\n"
]
}
],
"prompt_number": 21
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The sample in this example is balanced, all group levels have the same number of observations. We can drop some observations to get an unbalanced sample."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mc2s = multi.MultiComparison(dta2['StressReduction'][3:29], dta2['Treatment'][3:29])\n",
"res2s = mc2s.tukeyhsd()\n",
"print(res2s.summary())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Multiple Comparison of Means - Tukey HSD,FWER=0.05\n",
"============================================\n",
"group1 group2 meandiff lower upper reject\n",
"--------------------------------------------\n",
" 0 1 1.8889 0.6302 3.1476 True \n",
" 0 2 0.8889 -0.2587 2.0365 False \n",
" 1 2 -1.0 -2.2309 0.2309 False \n",
"--------------------------------------------\n"
]
}
],
"prompt_number": 22
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The test and confidence interval are by default defined for a familywise error rate of alpha=0.05. The FWER can be changed in the call to `tukeyhsd`. The confidence interval becomes larger if we reduce alpha to one percent, however in this case the rejection decision remains unchanged."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"res2s_001 = mc2s.tukeyhsd(alpha=0.01)\n",
"print(res2s_001.summary())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Multiple Comparison of Means - Tukey HSD,FWER=0.01\n",
"============================================\n",
"group1 group2 meandiff lower upper reject\n",
"--------------------------------------------\n",
" 0 1 1.8889 0.2662 3.5116 True \n",
" 0 2 0.8889 -0.5905 2.3683 False \n",
" 1 2 -1.0 -2.5868 0.5868 False \n",
"--------------------------------------------\n"
]
}
],
"prompt_number": 23
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from IPython.display import HTML\n",
"s = res2s_001.summary().as_html()\n",
"h = HTML(s); h"
],
"language": "python",
"metadata": {},
"outputs": [
{
"html": [
"<table class=\"simpletable\">\n",
"<caption>Multiple Comparison of Means - Tukey HSD,FWER=0.01</caption>\n",
"<tr>\n",
" <th>group1</th> <th>group2</th> <th>meandiff</th> <th>lower</th> <th>upper</th> <th>reject</th>\n",
"</tr>\n",
"<tr>\n",
" <td>0</td> <td>1</td> <td>1.8889</td> <td>0.2662</td> <td>3.5116</td> <td>True</td> \n",
"</tr>\n",
"<tr>\n",
" <td>0</td> <td>2</td> <td>0.8889</td> <td>-0.5905</td> <td>2.3683</td> <td>False</td>\n",
"</tr>\n",
"<tr>\n",
" <td>1</td> <td>2</td> <td>-1.0</td> <td>-2.5868</td> <td>0.5868</td> <td>False</td>\n",
"</tr>\n",
"</table>"
],
"output_type": "pyout",
"prompt_number": 24,
"text": [
"<IPython.core.display.HTML at 0x7739210>"
]
}
],
"prompt_number": 24
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"fig = res2.plot_simultaneous()\n",
"#Why are the axislabels shifted ?"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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7QwMul0tNmjRRRESEGjRooP79+6tt27a31AEAAACFkWXf5DsxEhMTFRMTc0sX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\nQmCC\n"
}
],
"prompt_number": 25
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 25
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 25
}
],
"metadata": {}
}
]
}
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