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By default, the columns are assumed to be arranged in order of increasing predicted mean." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "ranked", "annotation": "boolean, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "By default, " }, { "__type": "ParamRef", "__tag": 4071, "name": "data" }, { "__type": "Text", "__tag": 4046, "value": " is assumed to be observations rather than ranks; it will be ranked with " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.rankdata", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.stats._stats_py:rankdata" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " along " }, { "__type": "InlineCode", "__tag": 4051, "value": "axis=1" }, { "__type": "Text", "__tag": 4046, "value": ". If " }, { "__type": "ParamRef", "__tag": 4071, "name": "data" }, { "__type": "Text", "__tag": 4046, "value": " is provided in the form of ranks, pass argument " }, { "__type": "InlineCode", "__tag": 4051, "value": "True" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "predicted_ranks", "annotation": "array-like, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The predicted ranks of the column means. If not specified, the columns are assumed to be arranged in order of increasing predicted mean, so the default " }, { "__type": "ParamRef", "__tag": 4071, "name": "predicted_ranks" }, { "__type": "Text", "__tag": 4046, "value": " are " }, { "__type": "InlineMath", "__tag": 4057, "value": "[1, 2, \\dots, n-1, n]" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "{'auto', 'asymptotic', 'exact'}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Selects the method used to calculate the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "p" } ] }, { "__type": "Text", "__tag": 4046, "value": "-value. The following options are available." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'auto': selects between 'exact' and 'asymptotic' to achieve reasonably accurate results in reasonable time (default)" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'asymptotic': compares the standardized test statistic against the normal distribution" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'exact': computes the exact " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "p" } ] }, { "__type": "Text", "__tag": 4046, "value": "-value by comparing the observed " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " statistic against those realized by all possible permutations of ranks (under the null hypothesis that each permutation is equally likely)" } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Page's Test (also known as Page's " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " test) is useful when:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "there are " }, { "__type": "InlineMath", "__tag": 4057, "value": "n \\geq 3" }, { "__type": "Text", "__tag": 4046, "value": " treatments," } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineMath", "__tag": 4057, "value": "m \\geq 2" }, { "__type": "Text", "__tag": 4046, "value": " subjects are observed for each treatment, and" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "the observations are hypothesized to have a particular order." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Specifically, the test considers the null hypothesis that" } ] }, { "__type": "Math", "__tag": 4058, "value": "m_1 = m_2 = m_3 \\cdots = m_n," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "m_j" }, { "__type": "Text", "__tag": 4046, "value": " is the mean of the observed quantity under treatment " }, { "__type": "InlineMath", "__tag": 4057, "value": "j" }, { "__type": "Text", "__tag": 4046, "value": ", against the alternative hypothesis that" } ] }, { "__type": "Math", "__tag": 4058, "value": "m_1 \\leq m_2 \\leq m_3 \\leq \\cdots \\leq m_n," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where at least one inequality is strict." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "As noted by " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "4" }, { "__type": "Text", "__tag": 4046, "value": ", Page's " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " test has greater statistical power than the Friedman test against the alternative that there is a difference in trend, as Friedman's test only considers a difference in the means of the observations without considering their order. Whereas Spearman " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\rho" }, { "__type": "Text", "__tag": 4046, "value": " considers the correlation between the ranked observations of two variables (e.g. the airspeed velocity of a swallow vs. the weight of the coconut it carries), Page's " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " is concerned with a trend in an observation (e.g. the airspeed velocity of a swallow) across several distinct treatments (e.g. carrying each of five coconuts of different weight) even as the observation is repeated with multiple subjects (e.g. one European swallow and one African swallow)." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/stats/_page_trend_test.py", "item_line": 20, "item_type": "function", "aliases": [ "scipy.stats.page_trend_test" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "We use the example from [3]_: 10 students are asked to rate three\nteaching methods - tutorial, lecture, and seminar - on a scale of 1-5,\nwith 1 being the lowest and 5 being the highest. We have decided that\na confidence level of 99% is required to reject the null hypothesis in\nfavor of our alternative: that the seminar will have the highest ratings\nand the tutorial will have the lowest. Initially, the data have been\ntabulated with each row representing an individual student's ratings of\nthe three methods in the following order: tutorial, lecture, seminar.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "table = [[3, 4, 3],\n [2, 2, 4],\n [3, 3, 5],\n [1, 3, 2],\n [2, 3, 2],\n [2, 4, 5],\n [1, 2, 4],\n [3, 4, 4],\n [2, 4, 5],\n [1, 3, 4]]\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nBecause the tutorial is hypothesized to have the lowest ratings, the\ncolumn corresponding with tutorial rankings should be first; the seminar\nis hypothesized to have the highest ratings, so its column should be last.\nSince the columns are already arranged in this order of increasing\npredicted mean, we can pass the table directly into `page_trend_test`.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import page_trend_test\nres = page_trend_test(table)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThis *p*-value indicates that there is a 0.1819% chance that\nthe :math:`L` statistic would reach such an extreme value under the null\nhypothesis. Because 0.1819% is less than 1%, we have evidence to reject\nthe null hypothesis in favor of our alternative at a 99% confidence level.\n\nThe value of the :math:`L` statistic is 133.5. To check this manually,\nwe rank the data such that high scores correspond with high ranks, settling\nties with an average rank:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import rankdata\nranks = rankdata(table, axis=1)\nranks\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nWe add the ranks within each column, multiply the sums by the\npredicted ranks, and sum the products.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nm, n = ranks.shape\npredicted_ranks = np.arange(1, n+1)\nL = (predicted_ranks * np.sum(ranks, axis=0)).sum()\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res.statistic == L\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAs presented in [3]_, the asymptotic approximation of the *p*-value is the\nsurvival function of the normal distribution evaluated at the standardized\ntest statistic:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import norm\nE0 = (m*n*(n+1)**2)/4\nV0 = (m*n**2*(n+1)*(n**2-1))/144\nLambda = (L-E0)/np.sqrt(V0)\np = norm.sf(Lambda)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "p\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThis does not precisely match the *p*-value reported by `page_trend_test`\nabove. The asymptotic distribution is not very accurate, nor conservative,\nfor :math:`m \\leq 12` and :math:`n \\leq 8`, so `page_trend_test` chose to\nuse ``method='exact'`` based on the dimensions of the table and the\nrecommendations in Page's original paper [1]_. To override\n`page_trend_test`'s choice, provide the `method` argument.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = page_trend_test(table, method=\"asymptotic\")\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIf the data are already ranked, we can pass in the ``ranks`` instead of\nthe ``table`` to save computation time.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = page_trend_test(ranks, # ranks of data\n ranked=True, # data is already ranked\n )\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nSuppose the raw data had been tabulated in an order different from the\norder of predicted means, say lecture, seminar, tutorial.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "table = np.asarray(table)[:, [1, 2, 0]]\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nSince the arrangement of this table is not consistent with the assumed\nordering, we can either rearrange the table or provide the\n`predicted_ranks`. Remembering that the lecture is predicted\nto have the middle rank, the seminar the highest, and tutorial the lowest,\nwe pass:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = page_trend_test(table, # data as originally tabulated\n predicted_ranks=[2, 3, 1], # our predicted order\n )\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "friedmanchisquare", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "friedmanchisquare" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "rankdata", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "rankdata" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "spearmanr", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "spearmanr" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "data", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "ranked", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "predicted_ranks", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "auto" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "page_trend_test" }, "references": [ ".. [1] Ellis Batten Page, \"Ordered hypotheses for multiple treatments:", " a significant test for linear ranks\", *Journal of the American", " Statistical Association* 58(301), p. 216--230, 1963.", "", ".. [2] Markus Neuhauser, *Nonparametric Statistical Test: A computational", " approach*, CRC Press, p. 150--152, 2012.", "", ".. [3] Statext LLC, \"Page's L Trend Test - Easy Statistics\", *Statext -", " Statistics Study*, https://www.statext.com/practice/PageTrendTest03.php,", " Accessed July 12, 2020.", "", ".. [4] \"Page's Trend Test\", *Wikipedia*, WikimediaFoundation,", " https://en.wikipedia.org/wiki/Page%27s_trend_test,", " Accessed July 12, 2020.", "", ".. [5] Robert E. Odeh, \"The exact distribution of Page's L-statistic in", " the two-way layout\", *Communications in Statistics - Simulation and", " Computation*, 6(1), p. 49--61, 1977." ], "qa": "scipy.stats._page_trend_test:page_trend_test", "arbitrary": [], "local_refs": [ "data", "method", "predicted_ranks", "ranked", "res" ] }