{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Estimation and Hypothesis Testing" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "── Attaching packages ─────────────────────────────────────── tidyverse 1.2.1 ──\n", "✔ ggplot2 2.2.1 ✔ purrr 0.2.5\n", "✔ tibble 1.4.2 ✔ dplyr 0.7.5\n", "✔ tidyr 0.8.1 ✔ stringr 1.3.1\n", "✔ readr 1.1.1 ✔ forcats 0.3.0\n", "── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──\n", "✖ dplyr::filter() masks stats::filter()\n", "✖ dplyr::lag() masks stats::lag()\n" ] } ], "source": [ "library(tidyverse)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "options(repr.plot.width=4, repr.plot.height=3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Set random number seed for reproucibility" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "set.seed(42)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Functions around probability distributions" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Radnom numbers" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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"text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- seq(-3, 3, length.out = 100)\n", "plot(x, dnorm(x), type=\"l\", ylab=\"PDF\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "CDF " ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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WOtEqdm5zdtt25M/y664tq5wvwP1i7L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\nYOL8H8KfNlnf42rTAAAAAElFTkSuQmCC", "text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- seq(-3, 3, length.out = 100)\n", "plot(x, pnorm(x), type=\"l\", ylab=\"CDF\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Quantiles (inverse CDF)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p = seq(0, 1, length.out = 101)\n", "plot(p, qnorm(p), type=\"l\", ylab=\"Quantile\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Point estimates" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "x <- rnorm(10)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item -0.106124516091484\n", "\\item 1.51152199743894\n", "\\item -0.0946590384130976\n", "\\item 2.01842371387704\n", "\\item -0.062714099052421\n", "\\item 1.30486965422349\n", "\\item 2.28664539270111\n", "\\item -1.38886070111234\n", "\\item -0.278788766817371\n", "\\item -0.133321336393658\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. -0.106124516091484\n", "2. 1.51152199743894\n", "3. -0.0946590384130976\n", "4. 2.01842371387704\n", "5. -0.062714099052421\n", "6. 1.30486965422349\n", "7. 2.28664539270111\n", "8. -1.38886070111234\n", "9. -0.278788766817371\n", "10. -0.133321336393658\n", "\n", "\n" ], "text/plain": [ " [1] -0.10612452 1.51152200 -0.09465904 2.01842371 -0.06271410 1.30486965\n", " [7] 2.28664539 -1.38886070 -0.27878877 -0.13332134" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Mean" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Manual calculation" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.50569923003602" ], "text/latex": [ "0.50569923003602" ], "text/markdown": [ "0.50569923003602" ], "text/plain": [ "[1] 0.5056992" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sum(x)/length(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using built-in function" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.50569923003602" ], "text/latex": [ "0.50569923003602" ], "text/markdown": [ "0.50569923003602" ], "text/plain": [ "[1] 0.5056992" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mean(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Median" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Manual calculation" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "x_sorted <- sort(x)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "10" ], "text/latex": [ "10" ], "text/markdown": [ "10" ], "text/plain": [ "[1] 10" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "length(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since there are an even number of observations, we need the average of the middle two data poitns" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "-0.0786865687327593" ], "text/latex": [ "-0.0786865687327593" ], "text/markdown": [ "-0.0786865687327593" ], "text/plain": [ "[1] -0.07868657" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sum(x_sorted[5:6])/2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using built-in function" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "-0.0786865687327593" ], "text/latex": [ "-0.0786865687327593" ], "text/markdown": [ "-0.0786865687327593" ], "text/plain": [ "[1] -0.07868657" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "median(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Quantiles\n", "\n", "The mean is just the 50 percentile. We can use R to get any percentile we like." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "50%: -0.0786865687327593" ], "text/latex": [ "\\textbf{50\\textbackslash{}\\%:} -0.0786865687327593" ], "text/markdown": [ "**50%:** -0.0786865687327593" ], "text/plain": [ " 50% \n", "-0.07868657 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "quantile(x, 0.5)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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\n" ], "text/latex": [ "\\begin{description*}\n", "\\item[0\\textbackslash{}\\%] -1.38886070111234\n", "\\item[25\\textbackslash{}\\%] -0.126522131318115\n", "\\item[50\\textbackslash{}\\%] -0.0786865687327593\n", "\\item[75\\textbackslash{}\\%] 1.45985891163508\n", "\\item[100\\textbackslash{}\\%] 2.28664539270111\n", "\\end{description*}\n" ], "text/markdown": [ "0%\n", ": -1.3888607011123425%\n", ": -0.12652213131811550%\n", ": -0.078686568732759375%\n", ": 1.45985891163508100%\n", ": 2.28664539270111\n", "\n" ], "text/plain": [ " 0% 25% 50% 75% 100% \n", "-1.38886070 -0.12652213 -0.07868657 1.45985891 2.28664539 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "quantile(x, seq(0,1,length.out = 5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "Gene X is known to have a normal distribution with a mean of 100 units and a standard deviation of 15 units in the US population. With respect to this population," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- (1) What is the medan value for gene X?" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "100" ], "text/latex": [ "100" ], "text/markdown": [ "100" ], "text/plain": [ "[1] 100" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "100" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- (2) What is the probability of finding a value of more than 130 for gene X if you pick a person at random?" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "m <- 100\n", "s <- 15" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0227501319481792" ], "text/latex": [ "0.0227501319481792" ], "text/markdown": [ "0.0227501319481792" ], "text/plain": [ "[1] 0.02275013" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "1 - pnorm(130, m, s)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- (3) If you measrue gene X and find that it is in the 95th percentile for this population, what is the measured value?" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "124.672804404272" ], "text/latex": [ "124.672804404272" ], "text/markdown": [ "124.672804404272" ], "text/plain": [ "[1] 124.6728" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "qnorm(0.95, m, s)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- (4) Find answers to questions (1), (2) and (3) by simulating 1 million people sampled from the US population. Do they agree with the theoretical calculated values?" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "n <- 1e6\n", "x <- rnorm(n, m, s)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "100.020793858963" ], "text/latex": [ "100.020793858963" ], "text/markdown": [ "100.020793858963" ], "text/plain": [ "[1] 100.0208" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "median(x)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.022838" ], "text/latex": [ "0.022838" ], "text/markdown": [ "0.022838" ], "text/plain": [ "[1] 0.022838" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sum(x > 130)/n" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "95%: 124.695020504446" ], "text/latex": [ "\\textbf{95\\textbackslash{}\\%:} 124.695020504446" ], "text/markdown": [ "**95%:** 124.695020504446" ], "text/plain": [ " 95% \n", "124.695 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "quantile(x, 0.95)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- (5) Plot the PDF of gene X using `ggplot2` for values between 50 and 150. Give it a title of `PDF of N(100, 15)`, a subtile of `I made this!`, and label the x-axis as `Gene X` and y-axis as `PDF`. Make the PDF blue, and fill the region under the curve blue with a transparency of 50%." ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": {}, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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8/XarXWMmq1esaMGdabsbGxlZWVDoWHYZharWYYxmQyObSjK6jV6qqqKrGjQAqF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\nI0aYtVp+6VJlr17+v/3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"text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- 50:150\n", "df <- data.frame(x=x, y=dnorm(x, m, s))\n", "ggplot(df, aes(x=x, y=y)) +\n", "geom_line(color='blue') +\n", "geom_polygon(fill='blue', alpha=0.5) +\n", "labs(title='PDF of N(100, 15)',\n", " subtitle='I made this!',\n", " x='Gene X',\n", " y='PDF')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Interval estimates" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Confidence intervals" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [], "source": [ "ci = 0.95" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 94.215778757373
    2. \n", "\t
  2. 105.784221242627
    3. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 94.215778757373\n", "\\item 105.784221242627\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 94.215778757373\n", "2. 105.784221242627\n", "\n", "\n" ], "text/plain": [ "[1] 94.21578 105.78422" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "alpha = (1-ci)\n", "n <- length(x)\n", "m <- mean(x)\n", "s <- sd(x)\n", "se <- s/sqrt(n)\n", "me <- qt(1-alpha/2, df=n-1) * se\n", "c(m - me, m + me)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that confidence intervals get larger as the confidence required increases." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "ci = 0.99" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 92.3442793441823
    2. \n", "\t
  2. 107.655720655818
    3. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 92.3442793441823\n", "\\item 107.655720655818\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 92.3442793441823\n", "2. 107.655720655818\n", "\n", "\n" ], "text/plain": [ "[1] 92.34428 107.65572" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "alpha = (1-ci)\n", "n <- length(x)\n", "m <- mean(x)\n", "s <- sd(x)\n", "se <- s/sqrt(n)\n", "me <- qt(1-alpha/2, df=n-1) * se\n", "c(m - me, m + me)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Making a function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Review of R custom functions" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [], "source": [ "f <- function(a, b=1) {\n", " a + b\n", "}" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "3" ], "text/latex": [ "3" ], "text/markdown": [ "3" ], "text/plain": [ "[1] 3" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f(2)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "5" ], "text/latex": [ "5" ], "text/markdown": [ "5" ], "text/plain": [ "[1] 5" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f(2,3)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "5" ], "text/latex": [ "5" ], "text/markdown": [ "5" ], "text/plain": [ "[1] 5" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f(b=4, a=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "Make a function called `conf` for calculating confidence intervals for the sample mean that takes two arguments \n", "\n", "- x is the vector of sample values\n", "- ci is the confidence interval with a default of 0.95\n", "\n", "The funciton should return a vector of two numbers indicating the lwoer and upper limeit of the confidence interval\n", "\n", "Check that it gives the same answer as the example above." ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [], "source": [ "conf <- function(x, ci=0.95) {\n", " alpha = (1-ci)\n", " n <- length(x)\n", " m <- mean(x)\n", " s <- sd(x)\n", " se <- s/sqrt(n)\n", " me <- qt(1-alpha/2, df=n-1) * se\n", " c(m - me, m + me) \n", "}" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 94.215778757373
    2. \n", "\t
  2. 105.784221242627
    3. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 94.215778757373\n", "\\item 105.784221242627\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 94.215778757373\n", "2. 105.784221242627\n", "\n", "\n" ], "text/plain": [ "[1] 94.21578 105.78422" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "conf(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coverage\n", "\n", "In 1,000 experiments, we expect the true mean (0) to lie within the estimated 95% CIs 950 times." ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [], "source": [ "n_expt <- 1000\n", "n <- 10\n", "cls <- t(replicate(n_expt, conf(rnorm(n))))" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "960" ], "text/latex": [ "960" ], "text/markdown": [ "960" ], "text/plain": [ "[1] 960" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sum(cls[,1] < 0 & 0 < cls[,2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hypothesis testing" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Binomial test" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "tosses\n", " H T \n", "27 23 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "set.seed(123)\n", "\n", "n = 50\n", "tosses = sample(c('H', 'T'), n, replace=TRUE, prob=c(0.55, 0.45))\n", "t = table(tosses)\n", "t" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tExact binomial test\n", "\n", "data: table(tosses)\n", "number of successes = 27, number of trials = 50, p-value = 0.6718\n", "alternative hypothesis: true probability of success is not equal to 0.5\n", "95 percent confidence interval:\n", " 0.3932420 0.6818508\n", "sample estimates:\n", "probability of success \n", " 0.54 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "binom.test(table(tosses))" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "tosses\n", " H T \n", "139 111 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "set.seed(123)\n", "\n", "n = 250\n", "tosses = sample(c('H', 'T'), n, replace=TRUE, prob=c(0.55, 0.45))\n", "t = table(tosses)\n", "t" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tExact binomial test\n", "\n", "data: t\n", "number of successes = 139, number of trials = 250, p-value = 0.0875\n", "alternative hypothesis: true probability of success is not equal to 0.5\n", "95 percent confidence interval:\n", " 0.4920569 0.6185995\n", "sample estimates:\n", "probability of success \n", " 0.556 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "binom.test(t)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### What happens if we choose a one-sided test?" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tExact binomial test\n", "\n", "data: t\n", "number of successes = 139, number of trials = 250, p-value = 0.04375\n", "alternative hypothesis: true probability of success is greater than 0.5\n", "95 percent confidence interval:\n", " 0.5020197 1.0000000\n", "sample estimates:\n", "probability of success \n", " 0.556 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "binom.test(t, alternative = \"greater\")" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tExact binomial test\n", "\n", "data: t\n", "number of successes = 139, number of trials = 250, p-value = 0.9668\n", "alternative hypothesis: true probability of success is less than 0.5\n", "95 percent confidence interval:\n", " 0.0000000 0.6089811\n", "sample estimates:\n", "probability of success \n", " 0.556 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "binom.test(t, alternative = \"less\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### What happens if we change our null hypothesis?" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tExact binomial test\n", "\n", "data: t\n", "number of successes = 139, number of trials = 250, p-value = 0.8989\n", "alternative hypothesis: true probability of success is not equal to 0.55\n", "95 percent confidence interval:\n", " 0.4920569 0.6185995\n", "sample estimates:\n", "probability of success \n", " 0.556 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "binom.test(t, p = 0.55)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Two-sample model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Welch t-test" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "set.seed(123)\n", "\n", "n <- 10\n", "x1 <- rnorm(n, 0, 1)\n", "x2 <- rnorm(n, 1, 1)" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tWelch Two Sample t-test\n", "\n", "data: x1 and x2\n", "t = -2.5438, df = 17.872, p-value = 0.02044\n", "alternative hypothesis: true difference in means is not equal to 0\n", "95 percent confidence interval:\n", " -2.0710488 -0.1969438\n", "sample estimates:\n", " mean of x mean of y \n", "0.07462564 1.20862196 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t.test(x1, x2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Standard t-test" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tTwo Sample t-test\n", "\n", "data: x1 and x2\n", "t = -2.5438, df = 18, p-value = 0.02036\n", "alternative hypothesis: true difference in means is not equal to 0\n", "95 percent confidence interval:\n", " -2.0705694 -0.1974232\n", "sample estimates:\n", " mean of x mean of y \n", "0.07462564 1.20862196 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t.test(x1, x2, var.equal = TRUE)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Power of t-test" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [], "source": [ "d <- 1 # Effect size is ratio of difference in means to standard deviation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```R\n", "library(pwr)\n", "pwr.t.test(d = d, sig.level = 0.05, power = 0.9)\n", "```\n", "\n", "gives output\n", "\n", "```\n", " Two-sample t test power calculation \n", "\n", " n = 22.02109\n", " d = 1\n", " sig.level = 0.05\n", " power = 0.9\n", " alternative = two.sided\n", "\n", "NOTE: n is number in *each* group\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Interpretation of the power calculaiton\n", "\n", "If we did many experiments with `n=23` per group where the effect size is as specified and the test assumptions are valid, we expect that at least 90% of them will have a p-value less than the nominal significance level (0.05). If we used `n=22` we would expect that just under 90% of the experiments will have a p-value less than the nomial significance level (0.05).\n", "\n", "In particular notet that about 1-power of the experiments will fail to show a statistically significant p value even if the assumptionss are met (false negative)." ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.901" ], "text/latex": [ "0.901" ], "text/markdown": [ "0.901" ], "text/plain": [ "[1] 0.901" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "n_expts <- 10000\n", "n <- 22\n", "alpha = 0.05\n", "sum(replicate(n_expts, t.test(rnorm(n, 0, 1), rnorm(n, 1, 1))$p.value) < alpha)/n_expts" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.9125" ], "text/latex": [ "0.9125" ], "text/markdown": [ "0.9125" ], "text/plain": [ "[1] 0.9125" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "n_expts <- 10000\n", "n <- 23\n", "alpha = 0.05\n", "sum(replicate(n_expts, t.test(rnorm(n, 0, 1), rnorm(n, 1, 1))$p.value) < alpha)/n_expts" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Distribution of p-values under the null is uniform\n", "\n", "That meas that you expect $\\alpha$ of the experiments to be false positives." ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [], "source": [ "n_expt <- 10000\n", "n <- 50\n", "ps <- replicate(n_expts, t.test(rnorm(n), rnorm(n))$p.value)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0478" ], "text/latex": [ "0.0478" ], "text/markdown": [ "0.0478" ], "text/plain": [ "[1] 0.0478" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sum(ps < alpha)/n_expts" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "scrolled": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Plot with title “Histogram of ps”" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "hist(ps)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Paired and one-sample t-tests\n", "\n", "A paired t-test is commonly used to evaluate if paired measuremetns (e..g. weight before and after a diet for the same person) has changed. The paired t-test is equivalent to a one-sample t-test that compares the difference in measurements for the paired values with a fixed number (usually 0). " ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [], "source": [ "x1 <- rnorm(10, 100, 15)\n", "x2 <- rnorm(10, 100, 15)\n", "delta <- x1 - x2" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tPaired t-test\n", "\n", "data: x1 and x2\n", "t = -2.4618, df = 9, p-value = 0.03605\n", "alternative hypothesis: true difference in means is not equal to 0\n", "95 percent confidence interval:\n", " -12.3446069 -0.5215923\n", "sample estimates:\n", "mean of the differences \n", " -6.4331 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t.test(x1, x2, paired=TRUE)" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\tOne Sample t-test\n", "\n", "data: delta\n", "t = -2.4618, df = 9, p-value = 0.03605\n", "alternative hypothesis: true mean is not equal to 0\n", "95 percent confidence interval:\n", " -12.3446069 -0.5215923\n", "sample estimates:\n", "mean of x \n", " -6.4331 \n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t.test(delta, mu=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "Suppose that the null hypotehsis is that there is no difference in the paired measurements and the standard deviation of the differene is 2.\n", "\n", "- Run a simulation for 100,000 experiments with `n=25` per experiment to show the distributon of the p values using a paired or one -sample t-test under the null.\n", "- If the significance level is 0.05, how many false positive results were observed?" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [], "source": [ "n_expt <- 100000\n", "n <- 25\n", "s <- 2\n", "ps <- replicate(n_expts, t.test(rnorm(10, 0, s), mu=0)$p.value)" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "image/png": 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hUgwbYRnDGQyYLZxPbGChUgwbYRLN1V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", "text/plain": [ "Plot with title “Histogram of ps”" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "hist(ps)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "2.00293039283011" ], "text/latex": [ "2.00293039283011" ], "text/markdown": [ "2.00293039283011" ], "text/plain": [ "[1] 2.00293" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "n <- 100000\n", "s <- sqrt(2)\n", "sd(rnorm(n, 0, s) - rnorm(n, 0, s))" ] } ], "metadata": { "kernelspec": { "display_name": "R", "language": "R", "name": "ir" }, "language_info": { "codemirror_mode": "r", "file_extension": ".r", "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", "version": "3.4.0" } }, "nbformat": 4, "nbformat_minor": 2 }