{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import scipy as sp\n", "import scipy.stats\n", "import matplotlib.pyplot as plt\n", "import pandas as pd" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Concepts of Hypothesis Testing " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You all heard of **null hypothesis** and **alternative hypothesis**, depends on the evidences that we decide to reject the null hypothesis or not. However if we do not have evidences to reject null hypothesis, we can't say that we accept null hypothesis, rather we say that _we can't reject null hypothesis based on current information_.\n", "\n", "Sometimes you might encounter the term of **type I error** and **type II error**, the former characterises the probability of rejecting a true null hypothesis, the latter characterises the probability of failing to reject a false null hypothesis. It might sounds counter-intuitive at first sight, but the plot below tells all story. \n", "\n", "The higher the significance level the lower probability of having type I error, but it increases the probability of having type II error." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from plot_material import type12_error\n", "\n", "type12_error()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If you are yet bewildered, here is the guideline, the blue shaded area are genuinely generated by null distribution, however they are too distant (i.e. $2\\sigma$ away) from the mean ($0$ in this example), so they are mistakenly rejected, this is what we call _Type I Error_. \n", "\n", "The orange shaded area are actually generated by alternative distribution, however they are in the adjacent area of mean of null hypothesis, so we failed to reject they, but wrongly. And this is called _Type II Error_.\n", "\n", "As you can see from the chart, if null distribution and alternative are far away from each other, the probability of both type of errors diminish to trivial. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Rejection Region and p-Value" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Rejection region** is a range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favour of the alternative hypothesis.\n", "\n", "To put it another way, a value has to be far enough from the mean of null distribution to fall into rejection region, then the distance is the evidence that the value might not be produced by null distribution, therefore a rejection of null hypothesis.\n", "\n", "Let's use some real data for illustration. The data format is ```.csv```, best tool is ```pandas``` library." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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GenderHeightWeightIndex
0Male174964
1Male189872
2Female1851104
3Female1951043
4Male149613
\n", "
" ], "text/plain": [ " Gender Height Weight Index\n", "0 Male 174 96 4\n", "1 Male 189 87 2\n", "2 Female 185 110 4\n", "3 Female 195 104 3\n", "4 Male 149 61 3" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data = pd.read_csv(\"500_Person_Gender_Height_Weight_Index.csv\")\n", "data.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Null and alternative hypothesis are\n", "$$\n", "H_0: \\text{Average male height is 172}\\newline\n", "H_1: \\text{Average male height isn't 172}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Calculate the sample mean and standard deviation of male height" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "male_mean = data[data[\"Gender\"] == \"Male\"][\"Height\"].mean()\n", "male_std = data[data[\"Gender\"] == \"Male\"][\"Height\"].std(ddof=1)\n", "male_std_error = male_std / np.sqrt(len(data[data[\"Gender\"] == \"Male\"]))\n", "male_null = 172" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The rejection region is simply an opposite view of expressing confidence interval\n", "$$\n", "\\bar{x}>\\mu + t_\\alpha\\frac{s}{\\sqrt{n}}\\\\\n", "\\bar{x}<\\mu - t_\\alpha\\frac{s}{\\sqrt{n}}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Assume significance level $5\\%$, then $+t_\\alpha = t_{.025}$ and $-t_{\\alpha} = t_{.975}$, where $t_{.025}$ and $t_{.975}$ can be calculated by ```.stat.t.ppf```." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.9697339922715282" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = len(data[data[\"Gender\"] == \"Male\"]) - 1\n", "t_975 = sp.stats.t.ppf(0.975, df=df)\n", "t_975" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-1.9697339922715287" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t_025 = sp.stats.t.ppf(0.025, df=df)\n", "t_025" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The rejection region of null hypothesis is <169.85242784779035 and >174.14757215220965\n" ] } ], "source": [ "print(\n", " \"The rejection region of null hypothesis is <{} and >{}\".format(\n", " male_null - t_975 * male_std_error, male_null + t_975 * male_std_error\n", " )\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "whereas the ```male_mean``` falls into\n", "the rejection region, we reject null hypothesis in favour of alternative hypothesis" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "169.64897959183673" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "male_mean" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Alternatively we can construct $t$-statistic\n", "$$\n", "t=\\frac{\\bar{x}-\\mu}{s/\\sqrt{n}}\n", "$$\n", "Rejection region is where $t$-statistic larger or smaller than critical values\n", "$$\n", "t>t_{\\alpha} = t_{.025} \\text{ and } t