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jpcbertoldo/wilcoxon-demsar-zsplit-policy-test-analysis.ipynb

Last active January 4, 2023 13:33
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See this issue #17530 in scipy: https://github.com/scipy/scipy/issues/17530
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wilcoxon-demsar-zsplit-policy-test-analysis.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# modified `scipy.stats.wilcoxon`"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# @_rename_parameter(\"mode\", \"method\")\n",
"# @_axis_nan_policy_factory(\n",
"# wilcoxon_result_object, paired=True,\n",
"# n_samples=lambda kwds: 2 if kwds.get('y', None) is not None else 1,\n",
"# result_to_tuple=wilcoxon_result_unpacker, n_outputs=wilcoxon_outputs,\n",
"# )\n",
"def wilcoxon_modified(x, y=None, zero_method=\"wilcox\", correction=False,\n",
" alternative=\"two-sided\", method='auto', zero_diff_policy='original',):\n",
" \"\"\"Modified version of scipy.stats.wilcoxon to allow for ties in exact mode.\n",
"\n",
" The modifications are marked with comments.\n",
" \n",
" `zero_diff_policy` controls what to do with zero-difference cases.\n",
" \n",
" 1) `zero_diff_policy == 'original'` has the same behavior as `scipy.stats.wilcoxon`\n",
" \n",
" 2) `zero_diff_policy == 'demsar'`\n",
" \n",
" In the case where the number of zero-differences is odd, the function discards one of them, as suggested by [1].\n",
" \n",
" [1] J. Demšar, “Statistical Comparisons of Classifiers over Multiple Data Sets,” Journal of Machine Learning Research, vol. 7, no. 1, pp. 1–30, 2006. Url: http://jmlr.org/papers/v7/demsar06a.html \n",
" \n",
" The previous automatic switch from exact to approx mode is not used anymore.\n",
" \n",
" 3) `zero_diff_policy == 'donothing' does nothing, avoiding the automatic switch of mode 'exact' to 'approx' from the original function. \n",
" \n",
" \"\"\"\n",
" import warnings\n",
" import numpy as np\n",
" from scipy.stats._morestats import (WilcoxonResult, _axis_nan_policy_factory,\n",
" _get_wilcoxon_distr, _rename_parameter,\n",
" _stats_py, asarray, distributions,\n",
" find_repeats, sqrt, wilcoxon_outputs,\n",
" wilcoxon_result_object,\n",
" wilcoxon_result_unpacker)\n",
"\n",
" mode = method\n",
"\n",
" if mode not in [\"auto\", \"approx\", \"exact\"]:\n",
" raise ValueError(\"mode must be either 'auto', 'approx' or 'exact'\")\n",
"\n",
" if zero_method not in [\"wilcox\", \"pratt\", \"zsplit\"]:\n",
" raise ValueError(\"Zero method must be either 'wilcox' \"\n",
" \"or 'pratt' or 'zsplit'\")\n",
"\n",
" if alternative not in [\"two-sided\", \"less\", \"greater\"]:\n",
" raise ValueError(\"Alternative must be either 'two-sided', \"\n",
" \"'greater' or 'less'\")\n",
" \n",
" if zero_diff_policy not in [\"original\", \"demsar\", \"donothing\"]:\n",
" raise ValueError(\"zero_diff_policy must be 'original' or 'demsar'\")\n",
"\n",
" if y is None:\n",
" d = asarray(x)\n",
" if d.ndim > 1:\n",
" raise ValueError('Sample x must be one-dimensional.')\n",
" else:\n",
" x, y = map(asarray, (x, y))\n",
" if x.ndim > 1 or y.ndim > 1:\n",
" raise ValueError('Samples x and y must be one-dimensional.')\n",
" if len(x) != len(y):\n",
" raise ValueError('The samples x and y must have the same length.')\n",
" d = x - y\n",
"\n",
" if len(d) == 0:\n",
" res = WilcoxonResult(np.nan, np.nan)\n",
" if method == 'approx':\n",
" res.zstatistic = np.nan\n",
" return res\n",
"\n",
" if mode == \"auto\":\n",
" if len(d) <= 50:\n",
" mode = \"exact\"\n",
" else:\n",
" mode = \"approx\"\n",
"\n",
" n_zero = np.sum(d == 0)\n",
" \n",
" # ========================================================================\n",
" # ========================================================================\n",
" # ========================================================================\n",
" # Modification: allow for ties in exact mode\n",
" \n",
" if zero_diff_policy == 'original':\n",
" if n_zero > 0 and mode == \"exact\":\n",
" mode = \"approx\"\n",
" warnings.warn(\"Exact p-value calculation does not work if there are \"\n",
" \"zeros. Switching to normal approximation.\")\n",
" \n",
" elif zero_diff_policy == \"demsar\": \n",
" if n_zero % 2 == 1:\n",
" # discard one zero difference\n",
" # first [0] is for the tuple returned by np.where, \n",
" # second [0] is for the first element of the tuple\n",
" where_idx = np.where(d == 0)[0][0]\n",
" x = np.delete(x, where_idx)\n",
" if y is not None:\n",
" y = np.delete(y, where_idx)\n",
" d = np.delete(d, where_idx)\n",
" n_zero -= 1\n",
" \n",
" elif zero_diff_policy == \"donothing\":\n",
" # avoid the exact-to-approx switch\n",
" # zero_method == \"wilcox\" would be modified in some cases\n",
" pass\n",
" \n",
" else:\n",
" raise ValueError(f\"zero_diff_policy={zero_diff_policy}\")\n",
" # ========================================================================\n",
" # ========================================================================\n",
" # ========================================================================\n",
"\n",
" if mode == \"approx\":\n",
" if zero_method in [\"wilcox\", \"pratt\"]:\n",
" if n_zero == len(d):\n",
" raise ValueError(\"zero_method 'wilcox' and 'pratt' do not \"\n",
" \"work if x - y is zero for all elements.\")\n",
" if zero_method == \"wilcox\":\n",
" # Keep all non-zero differences\n",
" d = compress(np.not_equal(d, 0), d)\n",
"\n",
" count = len(d)\n",
" if count < 10 and mode == \"approx\":\n",
" warnings.warn(\"Sample size too small for normal approximation.\")\n",
"\n",
" r = _stats_py.rankdata(abs(d))\n",
" r_plus = np.sum((d > 0) * r)\n",
" r_minus = np.sum((d < 0) * r)\n",
"\n",
" if zero_method == \"zsplit\":\n",
" r_zero = np.sum((d == 0) * r)\n",
" r_plus += r_zero / 2.\n",
" r_minus += r_zero / 2.\n",
"\n",
" # return min for two-sided test, but r_plus for one-sided test\n",
" # the literature is not consistent here\n",
" # r_plus is more informative since r_plus + r_minus = count*(count+1)/2,\n",
" # i.e. the sum of the ranks, so r_minus and the min can be inferred\n",
" # (If alternative='pratt', r_plus + r_minus = count*(count+1)/2 - r_zero.)\n",
" # [3] uses the r_plus for the one-sided test, keep min for two-sided test\n",
" # to keep backwards compatibility\n",
" if alternative == \"two-sided\":\n",
" T = min(r_plus, r_minus)\n",
" else:\n",
" T = r_plus\n",
"\n",
" if mode == \"approx\":\n",
" mn = count * (count + 1.) * 0.25\n",
" se = count * (count + 1.) * (2. * count + 1.)\n",
"\n",
" if zero_method == \"pratt\":\n",
" r = r[d != 0]\n",
" # normal approximation needs to be adjusted, see Cureton (1967)\n",
" mn -= n_zero * (n_zero + 1.) * 0.25\n",
" se -= n_zero * (n_zero + 1.) * (2. * n_zero + 1.)\n",
"\n",
" replist, repnum = find_repeats(r)\n",
" if repnum.size != 0:\n",
" # Correction for repeated elements.\n",
" se -= 0.5 * (repnum * (repnum * repnum - 1)).sum()\n",
"\n",
" se = sqrt(se / 24)\n",
"\n",
" # apply continuity correction if applicable\n",
" d = 0\n",
" if correction:\n",
" if alternative == \"two-sided\":\n",
" d = 0.5 * np.sign(T - mn)\n",
" elif alternative == \"less\":\n",
" d = -0.5\n",
" else:\n",
" d = 0.5\n",
"\n",
" # compute statistic and p-value using normal approximation\n",
" z = (T - mn - d) / se\n",
" if alternative == \"two-sided\":\n",
" prob = 2. * distributions.norm.sf(abs(z))\n",
" elif alternative == \"greater\":\n",
" # large T = r_plus indicates x is greater than y; i.e.\n",
" # accept alternative in that case and return small p-value (sf)\n",
" prob = distributions.norm.sf(z)\n",
" else:\n",
" prob = distributions.norm.cdf(z)\n",
" elif mode == \"exact\":\n",
" # get pmf of the possible positive ranksums r_plus\n",
" pmf = _get_wilcoxon_distr(count)\n",
" # note: r_plus is int (ties not allowed), need int for slices below\n",
" r_plus = int(r_plus)\n",
" if alternative == \"two-sided\":\n",
" if r_plus == (len(pmf) - 1) // 2:\n",
" # r_plus is the center of the distribution.\n",
" prob = 1.0\n",
" else:\n",
" p_less = np.sum(pmf[:r_plus + 1])\n",
" p_greater = np.sum(pmf[r_plus:])\n",
" prob = 2*min(p_greater, p_less)\n",
" elif alternative == \"greater\":\n",
" prob = np.sum(pmf[r_plus:])\n",
" else:\n",
" prob = np.sum(pmf[:r_plus + 1])\n",
" prob = np.clip(prob, 0, 1)\n",
"\n",
" res = WilcoxonResult(T, prob)\n",
" if method == 'approx':\n",
" res.zstatistic = z\n",
" return res"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def wilcoxon_permutation(\n",
" x, \n",
" zero_method, \n",
" alternative, \n",
" rng, \n",
") -> float: \n",
" \"\"\"Wilcoxon signed-rank test with `scipy.stats.permutation_test` for p-value computation.\"\"\"\n",
" import scipy\n",
" import numpy as np\n",
" \n",
" def get_statistic(x_):\n",
" res = scipy.stats.wilcoxon(\n",
" x_,\n",
" # doesnt matter because it does not concern the statistic computation\n",
" correction=False,\n",
" # does not infere with the computation of the statistic\n",
" method=\"exact\", \n",
" # this one is passed on because it changes the \n",
" # statistic computation when zero_method == \"zsplit\"\n",
" zero_method=zero_method, \n",
" # must be greater to work with the permutation test\n",
" alternative='greater', \n",
" )\n",
" # res = wilcoxon_modified(\n",
" # x_,\n",
" # # doesnt matter because it does not concern the statistic computation\n",
" # correction=False,\n",
" # # does not infere with the computation of the statistic\n",
" # method=\"exact\", \n",
" # # avoid the exact-to-approx switch when\n",
" # # zero_method == \"wilcox\" \n",
" # # because the statistic would be modified when\n",
" # # n_zero > 0 \n",
" # zero_diff_policy=\"donothing\",\n",
" # # this one is passed on because it changes the \n",
" # # statistic computation when zero_method == \"zsplit\"\n",
" # zero_method=zero_method, \n",
" # # must be greater to work with the permutation test\n",
" # alternative='greater', \n",
" # )\n",
" return res.statistic\n",
" \n",
" return scipy.stats.permutation_test(\n",
" (x,), get_statistic, \n",
" vectorized=False, \n",
" permutation_type='samples', \n",
" random_state=rng, \n",
" n_resamples=np.inf,\n",
" alternative=alternative,\n",
" )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# tests"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def get_pvalues(\n",
" num_samples, \n",
" num_zeros, \n",
" num_ties, \n",
" seed,\n",
" alternative, \n",
" zero_method=\"zsplit\", # !!!! not wilcox as in `scipy.stats.wilcoxon`\n",
"):\n",
" import scipy, numpy\n",
" \n",
" rng = numpy.random.default_rng(seed)\n",
" x = rng.normal(size=(num_samples,))\n",
" \n",
" # add zeros\n",
" x[:num_zeros] = 0 \n",
" \n",
" # add ties\n",
" ties_start_idx = num_zeros\n",
" ties_end_idx = ties_start_idx + num_ties\n",
" x[ties_start_idx:ties_end_idx] = x[ties_start_idx] \n",
" \n",
" # get pvalues\n",
" res_original = scipy.stats.wilcoxon(\n",
" x, zero_method=zero_method, alternative=alternative,\n",
" correction=False, method='exact', \n",
" )\n",
" res_permutation = wilcoxon_permutation(x, zero_method=zero_method, alternative=alternative, rng=rng)\n",
" res_modified = wilcoxon_modified(\n",
" x, zero_method=zero_method, alternative=alternative,\n",
" zero_diff_policy='demsar',\n",
" correction=False, method='exact', \n",
" )\n",
" \n",
" return {\n",
" \"zero_method\": zero_method,\n",
" \"alternative\": alternative,\n",
" \"num_samples\": num_samples,\n",
" \"num_zeros\": num_zeros,\n",
" \"num_ties\": num_ties,\n",
" \"seed\": seed,\n",
" \"pvalue_original\": res_original.pvalue,\n",
" \"pvalue_permutation\": res_permutation.pvalue,\n",
" \"pvalue_modified\": res_modified.pvalue,\n",
" }"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Number of combinations: 88200\n"
]
}
],
"source": [
"from itertools import product\n",
"import random\n",
"import pandas as pd\n",
"\n",
"ALTERNATIVES = ['two-sided', 'greater', 'less']\n",
"ZERO_METHODS = [\"wilcox\", \"pratt\", \"zsplit\"]\n",
"\n",
"# all possible combinations of (num_samples, num_zeros, num_ties)\n",
"# for num_samples in range(3, 10)\n",
"NUMS = [\n",
" (num_samples, num_zeros, num_ties)\n",
" for num_samples in range(3, 10)\n",
" for num_zeros, num_ties in product(range(num_samples), range(num_samples))\n",
" if num_zeros + num_ties <= num_samples\n",
"]\n",
"SEEDS = list(range(50))\n",
"\n",
"# flatten the list of tuples because NUMS is a list of tuples\n",
"ALL_COMBINATIONS = [\n",
" (num_samples, num_zeros, num_ties, seed, alternative, zero_method)\n",
" for (num_samples, num_zeros, num_ties), seed, alternative, zero_method in product(NUMS, SEEDS, ALTERNATIVES, ZERO_METHODS, )\n",
"]\n",
"\n",
"# DEBUG \n",
"# ALL_COMBINATIONS = ALL_COMBINATIONS[:10]\n",
"\n",
"print(f\"Number of combinations: {len(ALL_COMBINATIONS)}\")"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Starting 20 engines with <class 'ipyparallel.cluster.launcher.LocalEngineSetLauncher'>\n"
]
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "bc2d1a298cb94f9ebb9088c535f39ea7",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
" 0%| | 0/20 [00:00<?, ?engine/s]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "a6fd00da0f6e4983a6e583e0d096f883",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"<lambda>: 0%| | 0/88200 [00:00<?, ?tasks/s]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Stopping engine(s): 1672765481\n",
"engine set stopped 1672765481: {'engines': {'0': {'exit_code': 0, 'pid': 3067495, 'identifier': '0'}, '1': {'exit_code': 0, 'pid': 3067512, 'identifier': '1'}, '2': {'exit_code': 0, 'pid': 3067526, 'identifier': '2'}, '3': {'exit_code': 0, 'pid': 3067546, 'identifier': '3'}, '4': {'exit_code': 0, 'pid': 3067563, 'identifier': '4'}, '5': {'exit_code': 0, 'pid': 3067593, 'identifier': '5'}, '6': {'exit_code': 0, 'pid': 3067624, 'identifier': '6'}, '7': {'exit_code': 0, 'pid': 3067665, 'identifier': '7'}, '8': {'exit_code': 0, 'pid': 3067706, 'identifier': '8'}, '9': {'exit_code': 0, 'pid': 3067722, 'identifier': '9'}, '10': {'exit_code': 0, 'pid': 3067767, 'identifier': '10'}, '11': {'exit_code': 0, 'pid': 3067783, 'identifier': '11'}, '12': {'exit_code': 0, 'pid': 3067832, 'identifier': '12'}, '13': {'exit_code': 0, 'pid': 3067892, 'identifier': '13'}, '14': {'exit_code': 0, 'pid': 3067909, 'identifier': '14'}, '15': {'exit_code': 0, 'pid': 3067979, 'identifier': '15'}, '16': {'exit_code': 0, 'pid': 3067991, 'identifier': '16'}, '17': {'exit_code': 0, 'pid': 3068055, 'identifier': '17'}, '18': {'exit_code': 0, 'pid': 3068077, 'identifier': '18'}, '19': {'exit_code': 0, 'pid': 3068120, 'identifier': '19'}}, 'exit_code': 0}\n",
"Stopping controller\n",
"Controller stopped: {'exit_code': 0, 'pid': 3067314, 'identifier': 'ipcontroller-1672765480-u0ek-3066868'}\n"
]
}
],
"source": [
"# SEQUENTIAL VERSION\n",
"\n",
"# from progressbar import progressbar\n",
"# # make the progress more homogeneous in terms of time\n",
"# random.shuffle(ALL_COMBINATIONS) \n",
"# records = [None] * len(ALL_COMBINATIONS) # preallocate\n",
"# for idx, args in progressbar(enumerate(ALL_COMBINATIONS), max_value=len(ALL_COMBINATIONS)):\n",
"# records[idx] = dict(**get_pvalues(*args))\n",
"\n",
"\n",
"# PARALLEL VERSION\n",
"\n",
"import ipyparallel as ipp\n",
"with ipp.Cluster() as rc:\n",
" \n",
" rc[:][\"wilcoxon_modified\"] = wilcoxon_modified\n",
" rc[:][\"wilcoxon_permutation\"] = wilcoxon_permutation\n",
" rc[:][\"get_pvalues\"] = get_pvalues\n",
" \n",
" view = rc.load_balanced_view()\n",
" asyncresult = view.map_async(\n",
" lambda args: get_pvalues(*args), ALL_COMBINATIONS, \n",
" # return_exceptions=True\n",
" )\n",
" asyncresult.wait_interactive()\n",
" records = asyncresult.get()"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"df = pd.DataFrame.from_records(records)\n",
"df.index.name = \"idx\"\n",
"df.to_csv(\"wilcoxon3.csv\")"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"\n",
"# analysis"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"# make a cell print all the outputs instead of just the last one\n",
"from IPython.core.interactiveshell import InteractiveShell\n",
"InteractiveShell.ast_node_interactivity = \"all\""
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"def savefig_and_display(fig, idx):\n",
" \"\"\"\n",
" some figs wouldnt show in the pdf otherwise\n",
" \"\"\"\n",
" fname = f\"fig_{idx:03}.svg\"\n",
" fig.write_image(fname)\n",
" from IPython.display import SVG, display\n",
" display(SVG(fname))"
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"# analysis 1: pvalue difference relative to the permutation method\n",
"\n",
"look at difference between the p-value given by a method and that of the permutation method\n",
"\n",
"0 is the ideal"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div><div id=529341ec-2320-41d2-a320-ba42be34b051 style=\"display:none; background-color:#9D6CFF; color:white; width:200px; height:30px; padding-left:5px; border-radius:4px; flex-direction:row; justify-content:space-around; align-items:center;\" onmouseover=\"this.style.backgroundColor='#BA9BF8'\" onmouseout=\"this.style.backgroundColor='#9D6CFF'\" onclick=\"window.commands?.execute('create-mitosheet-from-dataframe-output');\">See Full Dataframe in Mito</div> <script> if (window.commands?.hasCommand('create-mitosheet-from-dataframe-output')) document.getElementById('529341ec-2320-41d2-a320-ba42be34b051').style.display = 'flex' </script> <table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>zero_method</th>\n",
" <th>alternative</th>\n",
" <th>num_samples</th>\n",
" <th>num_zeros</th>\n",
" <th>num_ties</th>\n",
" <th>seed</th>\n",
" <th>method</th>\n",
" <th>pvalue</th>\n",
" <th>diff</th>\n",
" <th>absdiff</th>\n",
" <th>signdiff</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>pratt</td>\n",
" <td>greater</td>\n",
" <td>3</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>modified</td>\n",
" <td>0.375</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>pratt</td>\n",
" <td>greater</td>\n",
" <td>3</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>original</td>\n",
" <td>0.375</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>pratt</td>\n",
" <td>greater</td>\n",
" <td>3</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>1</td>\n",
" <td>modified</td>\n",
" <td>0.125</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>pratt</td>\n",
" <td>greater</td>\n",
" <td>3</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>1</td>\n",
" <td>original</td>\n",
" <td>0.125</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>pratt</td>\n",
" <td>greater</td>\n",
" <td>3</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>2</td>\n",
" <td>modified</td>\n",
" <td>0.875</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table></div>"
],
"text/plain": [
" zero_method alternative num_samples num_zeros num_ties seed method \\\n",
"0 pratt greater 3 0 0 0 modified \n",
"1 pratt greater 3 0 0 0 original \n",
"2 pratt greater 3 0 0 1 modified \n",
"3 pratt greater 3 0 0 1 original \n",
"4 pratt greater 3 0 0 2 modified \n",
"\n",
" pvalue diff absdiff signdiff \n",
"0 0.375 0.0 0.0 0.0 \n",
"1 0.375 0.0 0.0 0.0 \n",
"2 0.125 0.0 0.0 0.0 \n",
"3 0.125 0.0 0.0 0.0 \n",
"4 0.875 0.0 0.0 0.0 "
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import numpy as np\n",
"from pathlib import Path\n",
"import pandas as pd\n",
"\n",
"datacsv = Path('wilcoxon3.csv')\n",
"df = pd.read_csv(datacsv, index_col=0)\n",
"\n",
"# df.head(3)\n",
"\n",
"df = df.melt(id_vars=['zero_method', 'alternative', 'num_samples', 'num_zeros', 'num_ties', 'seed'], var_name=\"method\", value_name=\"pvalue\")\n",
"df[\"method\"] = df[\"method\"].apply(lambda s: s[len(\"pvalue_\"):])\n",
"\n",
"sort_columns = ['zero_method', 'alternative', 'num_samples', 'num_zeros', 'num_ties', 'seed', \"method\"]\n",
"df = df.sort_values(by=sort_columns).reset_index(drop=True)\n",
"\n",
"# df.head(3)\n",
"\n",
"mask_is_permutation = df[\"method\"] == \"permutation\"\n",
"\n",
"df_perm = df[mask_is_permutation]\n",
"df_perm = pd.concat([df_perm, df_perm], axis=0).sort_values(by=sort_columns).reset_index(drop=True)#.set_index(sort_columns)\n",
"\n",
"df = df[~mask_is_permutation].reset_index(drop=True)#.set_index(sort_columns)\n",
"\n",
"# df.head(3)\n",
"\n",
"df['diff'] = df['pvalue'] - df_perm['pvalue']\n",
"df['absdiff'] = abs(df['diff'])\n",
"df['signdiff'] = np.sign(df['diff'])\n",
"\n",
"df.head(5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## percentiles of abs diff"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
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"<div><div id=2f477edc-db6e-48e9-8f5f-6d688fe70a7b style=\"display:none; background-color:#9D6CFF; color:white; width:200px; height:30px; padding-left:5px; border-radius:4px; flex-direction:row; justify-content:space-around; align-items:center;\" onmouseover=\"this.style.backgroundColor='#BA9BF8'\" onmouseout=\"this.style.backgroundColor='#9D6CFF'\" onclick=\"window.commands?.execute('create-mitosheet-from-dataframe-output');\">See Full Dataframe in Mito</div> <script> if (window.commands?.hasCommand('create-mitosheet-from-dataframe-output')) document.getElementById('2f477edc-db6e-48e9-8f5f-6d688fe70a7b').style.display = 'flex' </script> <table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" <th>min</th>\n",
" <th>25%</th>\n",
" <th>50%</th>\n",
" <th>75%</th>\n",
" <th>max</th>\n",
" </tr>\n",
" <tr>\n",
" <th>zero_method</th>\n",
" <th>alternative</th>\n",
" <th>method</th>\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th rowspan=\"5\" valign=\"top\">pratt</th>\n",
" <th rowspan=\"2\" valign=\"top\">greater</th>\n",
" <th>modified</th>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>0.015625</td>\n",
" <td>0.085938</td>\n",
" <td>0.501953</td>\n",
" </tr>\n",
" <tr>\n",
" <th>original</th>\n",
" <td>0.0</td>\n",
" <td>0.006345</td>\n",
" <td>0.023345</td>\n",
" <td>0.066414</td>\n",
" <td>0.341345</td>\n",
" </tr>\n",
" <tr>\n",
" <th rowspan=\"2\" valign=\"top\">less</th>\n",
" <th>modified</th>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>0.046875</td>\n",
" <td>0.156250</td>\n",
" <td>0.935547</td>\n",
" </tr>\n",
" <tr>\n",
" <th>original</th>\n",
" <td>0.0</td>\n",
" <td>0.005859</td>\n",
" <td>0.022750</td>\n",
" <td>0.062052</td>\n",
" <td>0.341345</td>\n",
" </tr>\n",
" <tr>\n",
" <th>two-sided</th>\n",
" <th>modified</th>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>0.062500</td>\n",
" <td>0.218750</td>\n",
" <td>0.996094</td>\n",
" </tr>\n",
" <tr>\n",
" <th>...</th>\n",
" <th>...</th>\n",
" <th>...</th>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" </tr>\n",
" <tr>\n",
" <th rowspan=\"5\" valign=\"top\">zsplit</th>\n",
" <th>greater</th>\n",
" <th>original</th>\n",
" <td>0.0</td>\n",
" <td>0.006039</td>\n",
" <td>0.023248</td>\n",
" <td>0.062627</td>\n",
" <td>0.292892</td>\n",
" </tr>\n",
" <tr>\n",
" <th rowspan=\"2\" valign=\"top\">less</th>\n",
" <th>modified</th>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>0.019531</td>\n",
" <td>0.054688</td>\n",
" <td>0.289062</td>\n",
" </tr>\n",
" <tr>\n",
" <th>original</th>\n",
" <td>0.0</td>\n",
" <td>0.005859</td>\n",
" <td>0.021478</td>\n",
" <td>0.062500</td>\n",
" <td>0.292892</td>\n",
" </tr>\n",
" <tr>\n",
" <th rowspan=\"2\" valign=\"top\">two-sided</th>\n",
" <th>modified</th>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>0.031250</td>\n",
" <td>0.085938</td>\n",
" <td>0.500000</td>\n",
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" <tr>\n",
" <th>original</th>\n",
" <td>0.0</td>\n",
" <td>0.009704</td>\n",
" <td>0.047898</td>\n",
" <td>0.129355</td>\n",
" <td>0.585784</td>\n",
" </tr>\n",
" </tbody>\n",
"</table></div>"
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" min 25% 50% 75% max\n",
"zero_method alternative method \n",
"pratt greater modified 0.0 0.000000 0.015625 0.085938 0.501953\n",
" original 0.0 0.006345 0.023345 0.066414 0.341345\n",
" less modified 0.0 0.000000 0.046875 0.156250 0.935547\n",
" original 0.0 0.005859 0.022750 0.062052 0.341345\n",
" two-sided modified 0.0 0.000000 0.062500 0.218750 0.996094\n",
" original 0.0 0.011719 0.055436 0.162095 0.682689\n",
"wilcox greater modified 0.0 0.000000 0.003906 0.062500 0.501953\n",
" original 0.0 0.007055 0.030415 0.077360 0.341345\n",
" less modified 0.0 0.000000 0.031250 0.183594 0.935547\n",
" original 0.0 0.006011 0.029504 0.078510 0.341345\n",
" two-sided modified 0.0 0.000000 0.046875 0.242188 0.996094\n",
" original 0.0 0.008835 0.066218 0.163549 0.682689\n",
"zsplit greater modified 0.0 0.000000 0.015625 0.046875 0.285156\n",
" original 0.0 0.006039 0.023248 0.062627 0.292892\n",
" less modified 0.0 0.000000 0.019531 0.054688 0.289062\n",
" original 0.0 0.005859 0.021478 0.062500 0.292892\n",
" two-sided modified 0.0 0.000000 0.031250 0.085938 0.500000\n",
" original 0.0 0.009704 0.047898 0.129355 0.585784"
]
},
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shape-rendering=\"crispEdges\" width=\"95\" height=\"67\" x=\"0\" y=\"0\" style=\"stroke: rgb(68, 68, 68); stroke-opacity: 1; fill: rgb(255, 255, 255); fill-opacity: 1; stroke-width: 0px;\"/><g class=\"scrollbox\" transform=\"\" clip-path=\"url(#legendfacad1)\"><text class=\"legendtitletext\" text-anchor=\"start\" x=\"2\" y=\"18.2\" style=\"font-family: 'Open Sans', verdana, arial, sans-serif; font-size: 14px; fill: rgb(42, 63, 95); fill-opacity: 1; white-space: pre;\">method</text><g class=\"groups\" transform=\"\"><g class=\"traces\" transform=\"translate(0,32.7)\" style=\"opacity: 1;\"><text class=\"legendtext\" text-anchor=\"start\" x=\"40\" y=\"4.680000000000001\" style=\"font-family: 'Open Sans', verdana, arial, sans-serif; font-size: 12px; fill: rgb(42, 63, 95); fill-opacity: 1; white-space: pre;\">modified</text><g class=\"layers\" style=\"opacity: 1;\"><g class=\"legendfill\"/><g class=\"legendlines\"><path class=\"js-line\" d=\"M5,0h30\" style=\"fill: none; stroke: rgb(99, 110, 250); stroke-opacity: 1; stroke-width: 2px;\"/></g><g class=\"legendsymbols\"><g class=\"legendpoints\"/></g></g><rect class=\"legendtoggle\" x=\"0\" y=\"-9.5\" width=\"90\" height=\"19\" style=\"fill: rgb(0, 0, 0); fill-opacity: 0;\"/></g></g><g class=\"groups\" transform=\"\"><g class=\"traces\" transform=\"translate(0,51.7)\" style=\"opacity: 1;\"><text class=\"legendtext\" text-anchor=\"start\" x=\"40\" y=\"4.680000000000001\" style=\"font-family: 'Open Sans', verdana, arial, sans-serif; font-size: 12px; fill: rgb(42, 63, 95); fill-opacity: 1; white-space: pre;\">original</text><g class=\"layers\" style=\"opacity: 1;\"><g class=\"legendfill\"/><g class=\"legendlines\"><path class=\"js-line\" d=\"M5,0h30\" style=\"fill: none; stroke: rgb(239, 85, 59); stroke-opacity: 1; stroke-width: 2px;\"/></g><g class=\"legendsymbols\"><g class=\"legendpoints\"/></g></g><rect class=\"legendtoggle\" x=\"0\" y=\"-9.5\" width=\"90\" height=\"19\" style=\"fill: rgb(0, 0, 0); fill-opacity: 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}
],
"source": [
"dftmp = df.copy()\n",
"\n",
"descdf = dftmp\n",
"descdf = descdf.groupby(['zero_method', 'alternative', \"method\"])[\"absdiff\"].describe()\n",
"descdf = descdf[[\"min\", \"25%\", \"50%\", \"75%\", \"max\"]]\n",
"\n",
"descdf\n",
"\n",
"import plotly.express as px\n",
"fig = px.line(\n",
" descdf.reset_index().melt(id_vars=descdf.index.names, value_name=\"absdiff\", var_name=\"percentile\"), \n",
" x='percentile', y='absdiff', \n",
" line_group='method', \n",
" color='method', \n",
" facet_col=\"alternative\",\n",
" facet_row=\"zero_method\",\n",
" facet_row_spacing=0.05, \n",
" facet_col_spacing=0.02,\n",
").update_layout(\n",
" height=900,\n",
" width=900,\n",
" title = f\"absolute difference relative to the permutation method (lower is better)<br><br>\"\\\n",
" f\"<sup>per `aternative`, per `zero_method`</sup><br>\"\\\n",
" f\"<sup>percentile statistics all ('num_samples', 'num_zeros', 'num_ties', 'seed') confounded</sup><br>\",\n",
" margin=dict(t=200),\n",
").update_yaxes(matches=None, showticklabels=True, autorange=True)\n",
"\n",
"fig\n",
"savefig_and_display(fig, 0)"
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## `zero_method == \"zsplit\"`: percentiles of abs diff"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
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"<IPython.core.display.SVG object>"
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"output_type": "display_data"
}
],
"source": [
"dftmp = df.copy()\n",
"\n",
"zero_method = \"zsplit\"\n",
"mask = dftmp['zero_method'] == zero_method\n",
"dftmp = dftmp[mask]\n",
"\n",
"descdf = dftmp\n",
"descdf = descdf.groupby(['num_samples', 'alternative', \"method\"])[\"absdiff\"].describe()\n",
"descdf = descdf[[\"min\", \"25%\", \"50%\", \"75%\", \"max\"]]\n",
"descdf = descdf.reset_index().melt(id_vars=descdf.index.names, value_name=\"absdiff\", var_name=\"percentile\")\n",
"\n",
"import plotly.express as px\n",
"fig = px.line(\n",
" descdf, \n",
" x='percentile', y='absdiff', \n",
" line_group='method', \n",
" color='method', \n",
" facet_col=\"alternative\",\n",
" facet_row=\"num_samples\",\n",
" range_y=(0.00, 0.50),\n",
").update_layout(\n",
" height=1200,\n",
" width=1000,\n",
" title = f\"absolute difference relative to the permutation method (lower is better)<br><br>\"\\\n",
" f\"<sup>fixed `zero_method == 'zsplit'`, per `aternative`, per `num_samples` </sup><br>\"\\\n",
" f\"<sup>percentile statistics all ('num_zeros', 'num_ties', 'seed') confounded</sup><br>\",\n",
" margin=dict(t=180,b=0),\n",
").update_yaxes(\n",
")\n",
"\n",
"fig\n",
"savefig_and_display(fig, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## `zero_method == \"zsplit\"`: histogram sign of diff "
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
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"config": {
"plotlyServerURL": "https://plot.ly"
},
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"color": "#636efa",
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See comment scipy/scipy#17530 (comment)

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jpcbertoldo commented Jan 4, 2023 •
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update 2023-01-04

procedure

Obtain the p-value for the wilcoxon test using three methods:

  • (a) original: use the method in scipy.stats.wilcoxon (version scipy==1.9.3)
  • (b) modified: avoid the mode (or method) switch (from 'exact' to 'approx') and the policy to deal with zero differences the one suggested in [1]:
    • do nothing if the number of zeros is even
    • discard one if the number of zeros is odd
  • (c) permutation: use scipy.stats.wilcoxon to obtain the test statistic and obtain the p-value with scipy.stats.permutation_test.

(c) gives the expected p-value, so (a) and (b) should get as close to it as possible.

We generate (standard) gaussianly random 1D vectors (as if they were the wilcoxon differences, i.e. the function called with y=None) of variable size num_samples then force num_zeros entries to 0 and force num_ties entries to have the same number (but not zero). num_samples is in range(0, 10) and we consider all possible combinations of (num_zeros, num_ties) such that num_zeros + num_ties <= num_samples with 50 random seeds for each combination.

[1] J. Demšar, “Statistical Comparisons of Classifiers over Multiple Data Sets,” Journal of Machine Learning Research, vol. 7, no. 1, pp. 1–30, 2006. Url: http://jmlr.org/papers/v7/demsar06a.html

experiment variants

We test different configurations of the parameters alternative and zero_methods:

ALTERNATIVES = ['two-sided', 'greater', 'less']
ZERO_METHODS = ["wilcox", "pratt", "zsplit"]

analysis

We look at the differences between the considered methods (a) and (b) relative to method (c). We mostly look at the absolute differences (lower is better) and the sign of the difference (positive is better than negative). The subtitle of the plots in the pdf above explain each visualization.

results and conclusions

Please refer to the pdf above where the figures can be visualized (better with zoom in a pdf viewer), checck the notebook for details.

  1. (see sections 4.1, 5.1, and 5.2) the modification generally has larger abs error when zero_method == 'pratt' or zero_method == 'wilcox', but it improves it when zero_method == 'zsplit'
  2. (see section 4.2) the modification generally has lower abs error for all num_samples, and the improvement is bigger with lower num_samples
  3. (see section 4.3) same story for every num_samples, the modification:
    1. has more exact p-values (sign of diff is 0)
    2. avoids more false positives (positive diff sign is better because the estimated p-value is larger than the permutation)

Sections 4.4 and 4.5 show similar results in more detail, where one can see that improvements occur moslty when the number of num_zeros is high (relative to num_samples).

Section 5.3 confirms that the estimations are more often improved than degradated (positive "decrease of abs error"), and it's more relevant for lower num_samples).

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