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v2.1 of Ch13

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Jonathan Taylor
2023-08-20 19:37:25 -07:00
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Ch13-multiple-lab.Rmd
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---
jupyter:
jupytext:
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formats: ipynb,Rmd
main_language: python
text_representation:
extension: .Rmd
format_name: rmarkdown
format_version: '1.2'
jupytext_version: 1.14.7
---
# Chapter 13 # Chapter 13
@@ -245,6 +232,7 @@ mult_test(fund_mini_pvals, method = "holm", alpha=0.05)[:2]
As discussed previously, Manager One seems to perform particularly As discussed previously, Manager One seems to perform particularly
well, whereas Manager Two has poor performance. well, whereas Manager Two has poor performance.
```{python} ```{python}
fund_mini.mean() fund_mini.mean()

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Ch13-multiple-lab.ipynb
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"We include our usual imports seen in earlier labs." "We include our usual imports seen in earlier labs."
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"We also collect the new imports\n", "We also collect the new imports\n",
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"## Review of Hypothesis Tests\n", "## Review of Hypothesis Tests\n",
@@ -92,13 +92,13 @@
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"To begin, we use `ttest_1samp()` from the\n", "To begin, we use `ttest_1samp()` from the\n",
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"The $p$-value comes out to 0.931, which is not low enough to\n", "The $p$-value comes out to 0.931, which is not low enough to\n",
@@ -169,13 +169,13 @@
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"Since this is a simulated data set, we can create a $2 \\times 2$ table\n", "Since this is a simulated data set, we can create a $2 \\times 2$ table\n",
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"Therefore, at level $\\alpha=0.05$, we reject 15 of the 50 false\n", "Therefore, at level $\\alpha=0.05$, we reject 15 of the 50 false\n",
@@ -299,19 +299,20 @@
"amounts to quite a weak signal, and it resulted in a high number of\n", "amounts to quite a weak signal, and it resulted in a high number of\n",
"Type II errors. Let’s instead simulate data with a stronger signal,\n", "Type II errors. Let’s instead simulate data with a stronger signal,\n",
"so that the ratio of the mean to the standard deviation for the false\n", "so that the ratio of the mean to the standard deviation for the false\n",
"null hypotheses equals $1$. We make only 10 Type II errors.\n" "null hypotheses equals $1$. We make only 10 Type II errors.\n",
" "
] ]
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" " " "
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"## Family-Wise Error Rate\n", "## Family-Wise Error Rate\n",
@@ -417,13 +418,13 @@
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@@ -454,7 +455,7 @@
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"As discussed previously, even for moderate values of $m$ such as $50$,\n", "As discussed previously, even for moderate values of $m$ such as $50$,\n",
@@ -473,13 +474,13 @@
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@@ -505,7 +506,7 @@
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"The $p$-values are low for Managers One and Three, and high for the\n", "The $p$-values are low for Managers One and Three, and high for the\n",
@@ -530,7 +531,7 @@
}, },
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"The `mult_test()` function takes $p$-values and a `method` argument, as well as an optional\n", "The `mult_test()` function takes $p$-values and a `method` argument, as well as an optional\n",
@@ -541,13 +542,13 @@
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"The $p$-values `bonf` are simply the `fund_mini_pvalues` multiplied by 5 and truncated to be less than\n", "The $p$-values `bonf` are simply the `fund_mini_pvalues` multiplied by 5 and truncated to be less than\n",
@@ -580,13 +581,13 @@
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@@ -608,7 +609,7 @@
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"Therefore, using Bonferroni’s method, we are able to reject the null hypothesis only for Manager\n", "Therefore, using Bonferroni’s method, we are able to reject the null hypothesis only for Manager\n",
@@ -622,13 +623,13 @@
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"As discussed previously, Manager One seems to perform particularly\n", "As discussed previously, Manager One seems to perform particularly\n",
"well, whereas Manager Two has poor performance." "well, whereas Manager Two has poor performance.\n",
" "
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@@ -694,7 +696,7 @@
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"Is there evidence of a meaningful difference in performance between\n", "Is there evidence of a meaningful difference in performance between\n",
@@ -705,13 +707,13 @@
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@@ -733,7 +735,7 @@
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"The test results in a $p$-value of 0.038,\n", "The test results in a $p$-value of 0.038,\n",
@@ -757,13 +759,13 @@
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"The `pairwise_tukeyhsd()` function provides confidence intervals\n", "The `pairwise_tukeyhsd()` function provides confidence intervals\n",
@@ -817,13 +819,13 @@
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@@ -845,7 +847,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "6278d13c", "id": "8590f246",
"metadata": {}, "metadata": {},
"source": [ "source": [
"## False Discovery Rate\n", "## False Discovery Rate\n",
@@ -858,13 +860,13 @@
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"execution_count": 17, "execution_count": 17,
"id": "b5842190", "id": "2c88ec87",
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@@ -876,7 +878,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "80fc2fcc", "id": "80e77fab",
"metadata": {}, "metadata": {},
"source": [ "source": [
"There are far too many managers to consider trying to control the FWER.\n", "There are far too many managers to consider trying to control the FWER.\n",
@@ -887,13 +889,13 @@
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"id": "7c9d8bed", "id": "b6d56819",
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"execution": { "execution": {
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"shell.execute_reply": "2023-08-07T00:19:51.854196Z" "shell.execute_reply": "2023-08-21T02:29:19.010371Z"
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"outputs": [ "outputs": [
@@ -916,7 +918,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "4f73096d", "id": "b4662444",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The *q-values* output by the\n", "The *q-values* output by the\n",
@@ -932,13 +934,13 @@
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"execution_count": 19, "execution_count": 19,
"id": "bfa39f7c", "id": "b00da3a1",
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"execution": { "execution": {
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@@ -960,7 +962,7 @@
}, },
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"cell_type": "markdown", "cell_type": "markdown",
"id": "ccb44c8d", "id": "fdccb808",
"metadata": {}, "metadata": {},
"source": [ "source": [
"We find that 146 of the 2,000 fund managers have a $q$-value below\n", "We find that 146 of the 2,000 fund managers have a $q$-value below\n",
@@ -976,13 +978,13 @@
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"execution_count": 20, "execution_count": 20,
"id": "70b69b47", "id": "1c230117",
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@@ -1004,7 +1006,7 @@
}, },
{ {
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"id": "c8a969f4", "id": "6112239d",
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"source": [ "source": [
"Figure 13.6 displays the ordered\n", "Figure 13.6 displays the ordered\n",
@@ -1026,13 +1028,13 @@
{ {
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"execution_count": 21, "execution_count": 21,
"id": "4c0ddea1", "id": "62289650",
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"execution": { "execution": {
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"shell.execute_reply": "2023-08-07T00:19:51.868357Z" "shell.execute_reply": "2023-08-21T02:29:19.023338Z"
} }
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@@ -1051,7 +1053,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "ddeb3900", "id": "c36b13b7",
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"source": [ "source": [
"We now reproduce the middle panel of Figure 13.6." "We now reproduce the middle panel of Figure 13.6."
@@ -1060,13 +1062,13 @@
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@@ -1096,7 +1098,7 @@
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"source": [ "source": [
"## A Re-Sampling Approach\n", "## A Re-Sampling Approach\n",
@@ -1110,13 +1112,13 @@
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@@ -1145,7 +1147,7 @@
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"metadata": {}, "metadata": {},
"source": [ "source": [
"There are four classes of cancer. For each gene, we compare the mean\n", "There are four classes of cancer. For each gene, we compare the mean\n",
@@ -1161,13 +1163,13 @@
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"execution_count": 24, "execution_count": 24,
"id": "96fb2f61", "id": "1afbcf47",
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@@ -1195,7 +1197,7 @@
}, },
{ {
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"id": "3131124e", "id": "61f24919",
"metadata": {}, "metadata": {},
"source": [ "source": [
"However, this $p$-value relies on the assumption that under the null\n", "However, this $p$-value relies on the assumption that under the null\n",
@@ -1214,13 +1216,13 @@
{ {
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"execution_count": 25, "execution_count": 25,
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@@ -1253,7 +1255,7 @@
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{ {
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"id": "c7fc4557", "id": "a97f74af",
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"source": [ "source": [
"This fraction, 0.0398,\n", "This fraction, 0.0398,\n",
@@ -1265,13 +1267,13 @@
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"execution_count": 26, "execution_count": 26,
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@@ -1307,7 +1309,7 @@
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"metadata": {}, "metadata": {},
"source": [ "source": [
"The re-sampling-based null distribution is almost identical to the theoretical null distribution, which is displayed in red.\n", "The re-sampling-based null distribution is almost identical to the theoretical null distribution, which is displayed in red.\n",
@@ -1325,13 +1327,13 @@
{ {
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"execution_count": 27, "execution_count": 27,
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@@ -1358,7 +1360,7 @@
}, },
{ {
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"id": "1b92df1b", "id": "06286699",
"metadata": {}, "metadata": {},
"source": [ "source": [
"Next, we compute the number of rejected null hypotheses $R$, the\n", "Next, we compute the number of rejected null hypotheses $R$, the\n",
@@ -1371,13 +1373,13 @@
{ {
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"execution_count": 28, "execution_count": 28,
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@@ -1394,7 +1396,7 @@
}, },
{ {
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"id": "f6779ea0", "id": "e26b64c6",
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"source": [ "source": [
"Now, for any given FDR, we can find the genes that will be\n", "Now, for any given FDR, we can find the genes that will be\n",
@@ -1412,13 +1414,13 @@
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"execution_count": 29, "execution_count": 29,
"id": "9661eb10", "id": "f11339e5",
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@@ -1457,7 +1459,7 @@
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{ {
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"At an FDR threshold of 0.2, more genes are selected, at the cost of having a higher expected\n", "At an FDR threshold of 0.2, more genes are selected, at the cost of having a higher expected\n",
@@ -1467,13 +1469,13 @@
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@@ -1524,7 +1526,7 @@
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"The next line generates Figure 13.11, which is similar\n", "The next line generates Figure 13.11, which is similar\n",
@@ -1535,13 +1537,13 @@
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@@ -1566,18 +1568,18 @@
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"\n" " \n"
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