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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Projekt - Test t studenta\n",
"\n",
"- Marcin Kostrzewski\n",
"- Krystian Wasilewski\n",
"- Mateusz Tylka\n",
"\n",
"## Test t studenta\n",
"\n",
"Metoda statystyczna służącą do porównania dwóch średnich między sobą gdy znamy liczbę badanych próbek, średnią arytmetyczną oraz wartość odchylenia standardowego lub wariancji.\n",
"Jest to jeden z mniej skomplikowanych i bardzo często wykorzystywanych testów statystycznych używanych do weryfikacji hipotez. Dzięki niemu możemy dowiedzieć się czy dwie różne średnie są różne niechcący (w wyniku przypadku) czy są różne istotnie statystycznie (np. z uwagi na naszą manipulację eksperymentalna).\n",
"Wyróżniamy 3 wersję testu t:\n",
"\n",
"1. test t Studenta dla jednej próby\n",
"2. test t Studenta dla prób niezależnych \n",
"3. test t Studenta dla prób zależnych"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Test Shapiro Wilka\n",
"\n",
"Wszystkie rodzaje testów są testami parametrycznymi, a co za tym idzie nasze mierzone zmienne ilościowe powinny mieć rozkład normalny. \n",
"Dzięki testowi Shapiro Wilka możemy sprawdzić to założenie."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Testowanie hipotez metodą bootstrap\n",
"\n",
"**Bootstrap** – metoda szacowania (estymacji) wyników poprzez wielokrotne losowanie ze zwracaniem z próby. Polega ona na utworzeniu nowego rozkładu wyników, na podstawie posiadanych danych, poprzez wielokrotne losowanie wartości z posiadanej próby. Metoda ze zwracaniem polega na tym, że po wylosowaniu danej wartości, “wraca” ona z powrotem do zbioru.\n",
"\n",
"Metoda bootstrapowa znajduje zastosowanie w sytuacji, w której nie znamy rozkładu z populacji z której pochodzi próbka lub w przypadku rozkładów małych lub asymetrycznych. W takim wypadku, dzięki tej metodzie, wyniki testów parametrycznych i analiz opartych o modele liniowe są bardziej precyzyjne. Zazwyczaj losuje się wiele próbek, np. 2000 czy 5000."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Definicje funkcji"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"from enum import Enum\n",
"from scipy.stats import ttest_ind, ttest_1samp, ttest_rel, shapiro"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [],
"source": [
"dataset = pd.read_csv('experiment_data.csv') # TODO: del?"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [],
"source": [
"class Alternatives(Enum):\n",
" LESS = 'less'\n",
" GREATER = 'greater'"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [],
"source": [
"def calculate_t_difference(t_stat_sample, t_stat_list, alternative):\n",
" \"\"\"\n",
" Funkcja oblicza procent statystyk testowych powstałych z prób bootstrapowych, \n",
" które róznią się od statystyki testowej powstałej ze zbioru według hipotezy alternatywnej.\n",
" \"\"\"\n",
" all_stats = len(t_stat_list)\n",
" stats_different_count = 0\n",
" for t_stat_boot in t_stat_list:\n",
" if alternative is Alternatives.LESS and t_stat_boot > t_stat_sample:\n",
" stats_different_count += 1 \n",
" elif alternative is Alternatives.GREATER and t_stat_boot < t_stat_sample:\n",
" stats_different_count += 1\n",
" return stats_different_count / all_stats"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def t_test_1_samp(sample_1, population_mean=None, alternative=Alternatives.LESS):\n",
" \"\"\"\n",
" Funkcja przeprowadza test T-studenta dla jednej zmiennej.\n",
" \"\"\"\n",
" t_stat_from_sample, _ = ttest_1samp(a=sample_1, popmean=population_mean, alternative=alternative.value)\n",
" t_stat_list = get_t_stats(sample_1, t_stat_fn=ttest_1samp, alternative=alternative, population_mean=population_mean)\n",
"\n",
" p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
"\n",
" return p, t_stat_from_sample, t_stat_list"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"def t_test_ind(sample_1, sample_2, alternative=Alternatives.LESS):\n",
" \"\"\"\n",
" Funkcja przeprowadza test T-studenta dla dwóch zmiennych niezależnych.\n",
" \"\"\"\n",
" t_stat_from_sample, _ = ttest_ind(sample_1, sample_2, alternative=alternative.value)\n",
" t_stat_list = get_t_stats(sample_1, sample_2, alternative=alternative, t_stat_fn=ttest_ind)\n",
"\n",
" p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
"\n",
" return p, t_stat_from_sample, t_stat_list"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [],
"source": [
"def t_test_dep(sample_1, sample_2, alternative=Alternatives.LESS):\n",
" \"\"\"\n",
" Funkcja przeprowadza test T-studenta dla dwóch zmiennych zależnych.\n",
" \"\"\"\n",
" t_stat_list = get_t_stats(sample_1, sample_2, alternative=alternative, t_stat_fn=ttest_rel)\n",
" t_stat_from_sample, _ = ttest_rel(sample_1, sample_2, alternative=alternative.value)\n",
"\n",
" p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
"\n",
" return p, t_stat_from_sample, t_stat_list"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
"def get_t_stats(sample_1, sample_2=None, t_stat_fn=ttest_1samp, alternative=Alternatives.LESS, population_mean=None):\n",
" \"\"\"Funkcja oblicza listę statystyk testowych dla każdej próbki bootstrapowej wybranej na podstawie danych sample_1 i sample_2\"\"\"\n",
" t_stat_list = []\n",
"\n",
" # One sample test\n",
" if t_stat_fn is ttest_1samp and sample_2 is None:\n",
" if not population_mean:\n",
" raise Exception(\"population_mean not provided\")\n",
" for bootstrap in generate_bootstraps(sample_1):\n",
" stat, _ = t_stat_fn(bootstrap, population_mean, alternative=alternative.value)\n",
" t_stat_list.append(stat)\n",
" return t_stat_list\n",
"\n",
" # Two sample test\n",
" for bootstrap_sample in generate_bootstraps(pd.concat((sample_1, sample_2), ignore_index=True)):\n",
" bootstrap_1 = bootstrap_sample.iloc[: len(bootstrap_sample) // 2]\n",
" bootstrap_2 = bootstrap_sample.iloc[len(bootstrap_sample) // 2 :]\n",
" stat, _ = t_stat_fn(bootstrap_1, bootstrap_2, alternative=alternative.value)\n",
" t_stat_list.append(stat)\n",
" return t_stat_list"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [],
"source": [
"def pretty_print_test(p, t_stat_from_sample, t_stat_list, thesis, alternative, max_print=5):\n",
" print('Wyniki bootstrapowej wersji testu T-studenta')\n",
" print()\n",
" print(f'Hipoteza: {thesis}')\n",
" if alternative is Alternatives.LESS:\n",
" print(f'Hipoteza alternatywna: średnia jest mniejsza')\n",
" else:\n",
" print(f'Hipoteza alternatywna: średnia jest większa')\n",
" print()\n",
" print(f'p: {p}')\n",
" print(f'Wartość statystyki testowej z próby: {t_stat_from_sample}')\n",
" print(f'Wartości statystyk z prób boostrapowych:')\n",
"\n",
" t_stat_list_len = len(t_stat_list)\n",
" for i in range(min(max_print, t_stat_list_len)):\n",
" print(f'{t_stat_list[i]}, ', end='')\n",
" if max_print < t_stat_list_len:\n",
" remaining = t_stat_list_len - max_print\n",
" print(f'... (i {remaining} pozostałych)')\n",
"\n",
" print()\n",
" print()"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def generate_bootstraps(data, n_bootstraps=1000):\n",
" data_size = data.shape[0]\n",
" for _ in range(n_bootstraps):\n",
" indices = np.random.choice(len(data), size=data_size)\n",
" yield data.iloc[indices, :]"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def bootstrap_one_sample(sample, population_mean, alternative=Alternatives.LESS):\n",
" p, t, ts = t_test_1_samp(\n",
" sample_1=sample,\n",
" population_mean=population_mean,\n",
" alternative=alternative,\n",
" )\n",
" \n",
" pretty_print_test(p, t, ts, f'średnia jest równa {population_mean}', alternative)\n",
" print()\n",
" return p, t, ts"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def bootstrap_independent(sample_1, sample_2, alternative=Alternatives.LESS):\n",
" p, t, ts = t_test_ind(\n",
" sample_1=sample_1,\n",
" sample_2=sample_2,\n",
" alternative=alternative,\n",
" )\n",
" \n",
" pretty_print_test(p, t, ts, 'średnie są takie same', alternative)\n",
" return p, t, ts"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def bootstrap_dependent(sample_1, sample_2, alternative=Alternatives.LESS):\n",
" p, t, ts = t_test_dep(\n",
" sample_1=sample_1,\n",
" sample_2=sample_2,\n",
" alternative=alternative,\n",
" )\n",
" \n",
" pretty_print_test(p, t, ts, 'średnie są takie same', alternative)\n",
" return p, t, ts"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def draw_distribution(stats, comparision_value):\n",
" \"\"\"\n",
" Funkcja rysuje rozkład statystyki testowej\n",
" @param stats: lista statystyk testowych\n",
" @param comparision_value: pierwotna próbka\n",
" \"\"\"\n",
" plt.hist(stats)\n",
" plt.axvline(comparision_value, color='red')\n",
" plt.xlabel('Test statistic value')\n",
" plt.ylabel('Frequency')\n",
" plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Wczytanie danych"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"0 169.5557\ndtype: float64\n0 175.1417\ndtype: float64\n0 79.6342\ndtype: float64\n0 76.5602\ndtype: float64\n"
]
}
],
"source": [
"dataset = pd.read_csv('experiment_data.csv')\n",
"heights_female = pd.DataFrame(dataset['Female height'].to_numpy()) # xd\n",
"heights_male = pd.DataFrame(dataset['Male height'].to_numpy())\n",
"weights_before = pd.DataFrame(dataset['Weight before'].to_numpy())\n",
"weights_after = pd.DataFrame(dataset['Weight after'].to_numpy())\n",
"print(np.mean(heights_female))\n",
"print(np.mean(heights_male))\n",
"print(np.mean(weights_before))\n",
"print(np.mean(weights_after))\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Jedna próba\n",
"\n",
"**Test t Studenta dla jednej próby** wykorzystujemy gdy chcemy porównać średnią “teoretyczną” ze średnią, którą faktycznie możemy zaobserwować w naszej bazie danych. Średnia teoretyczna to średnia pochodząca z innych badań lub po prostu bez większych uzasadnień pochodząca z naszej głowy.\n",
"\n",
"Wyobraźmy sobie, że mamy dane z takimi zmiennymi jak wzrost pewnej grupy ludzi. Dzięki testowi t Studenta dla jednej próby możemy dowiedzieć się np. czy wzrost naszego młodszego brata wynoszący 160cm odbiega znacząco od średniej wzrostu tej grupy.\n",
"\n",
"### Hipoteza\n",
"\n",
"*H0: Badana próba została wylosowana z populacji, w której wzrost osób wynosi średnio 160cm.* \n",
"*H1: Badana próba została wylosowana z populacji gdzie średni wzrost jest większy 160cm.*\n",
"\n",
"### Sprawdzenie założeń\n"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"p = 0.791\n"
]
}
],
"source": [
"# Sprawdzamy, czy próby mają rozkład normalny\n",
"shapiro_test = shapiro(heights_female)\n",
"print(f\"p = {round(shapiro_test.pvalue,4)}\")"
]
},
{
"source": [
"P wartość jest większa niż alfa = 0.05, więc próba ma prawdopodobnie rozkład normalny. Możemy stostować testy."
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Test\n"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnia jest równa 160.0\nHipoteza alternatywna: średnia jest większa\n\np: 0.5\nWartość statystyki testowej z próby: [19.1207964]\nWartości statystyk z prób boostrapowych:\n[17.41702865], [19.17874674], [20.59090525], [17.666445], [19.3593138], ... (i 95 pozostałych)\n\n\n\n"
]
},
{
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"tested_mean = 160.0\n",
"\n",
"p, t, ts = bootstrap_one_sample(heights_female, tested_mean, alternative=Alternatives.GREATER)\n",
"draw_distribution([x[0] for x in ts], t)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Wniosek\n",
"\n",
"Nie mamy podstaw, żeby odrzucić hipotezę zerową mówiącą, że średnia wynosi 160."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Dwie próby niezależne\n",
"\n",
"**Test t Studenta dla prób niezależnych** jest najczęściej stosowaną metodą statystyczną w celu porównania średnich z dwóch niezależnych od siebie grup. Wykorzystujemy go gdy chcemy porównać dwie grupy pod względem jakiejś zmiennej ilościowej. Na przykład gdy chcemy porównać średni wzrost kobiet i mężczyzn w danej grupie.\n",
"Zazwyczaj dwie średnie z różnych od siebie grup będą się różnić. Test t Studenta powie nam jednak czy owe różnice są istotne statystycznie – czy nie są przypadkowe.\n",
"Jeśli wynik testu t Studenta będzie istotny na poziomie p < 0,05 możemy odrzucić hipotezę zerową na rzecz hipotezy alternatywnej.\n",
"\n",
"## Hipoteza\n",
"\n",
"*H0: Średni wzrost w grupie mężczyzn jest taki sam jak średni w grupie kobiet. Hipoteza alternatywna z kolei* \n",
"*H1: Kobiety będą niższe od mężczyzn pod względem wzrostu.*\n",
"\n",
"## Sprawdzenie założeń\n",
"\n",
"Założenie o rozkładzie normalnym danych - sprawdzane testem Shapiro-Wilka"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"p = 0.791\np = 0.7535\n"
]
}
],
"source": [
"shapiro_test = shapiro(heights_female)\n",
"print(f\"p = {round(shapiro_test.pvalue,4)}\")\n",
"\n",
"shapiro_test = shapiro(heights_male)\n",
"print(f\"p = {round(shapiro_test.pvalue,4)}\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Wartości **p** w teście Shapiro-Wilka powyżej **0.05** -> Dane prawdopodobnie mają rozkład normalny"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Test"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnie są takie same\nHipoteza alternatywna: średnia jest mniejsza\n\np: 0.0\nWartość statystyki testowej z próby: [8.04931557]\nWartości statystyk z prób boostrapowych:\n[0.2748409], [-0.61193473], [1.24335163], [-2.56879464], [0.34249038], ... (i 95 pozostałych)\n\n\n"
]
},
{
"output_type": "display_data",
"data": {
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"p, t, ts = bootstrap_independent(heights_male, heights_female)\n",
"ts = [x[0] for x in ts]\n",
"draw_distribution(ts, t)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Wniosek"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Dwie próby zależne\n",
"\n",
"W odróżnieniu od testu dla prób niezależnych, gdzie porównujemy dwie grupy, ten rodzaj testu stosujemy gdy poddajemy analizie tą samą pojedynczą grupę, ale dwukrotnie w czasie.\n",
"\n",
"**Przykład**: Porównane zostały wagi przed dietą i po diecie.\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Hipoteza\n",
"H0 - Średnia waga nie uległa zmianie po zastosowaniu diety\n",
"H1 - Średnia waga po diecie jest znacząco mniejsza od wagi przed dietą\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"\n",
"### Sprawdzenie założeń\n",
"\n",
"Założenie o rozkładzie normalnym danych - sprawdzane testem Shapiro-Wilka"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"p = 0.3308\np = 0.4569\n"
]
}
],
"source": [
"shapiro_test = shapiro(weights_before)\n",
"print(f\"p = {round(shapiro_test.pvalue,4)}\")\n",
"\n",
"shapiro_test = shapiro(weights_after)\n",
"print(f\"p = {round(shapiro_test.pvalue,4)}\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Wartości **p** w teście Shapiro-Wilka powyżej **0.05** -> Dane prawdopodobnie mają rozkład normalny"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Test"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnie są takie same\nHipoteza alternatywna: średnia jest mniejsza\n\np: 0.0\nWartość statystyki testowej z próby: [48.30834167]\nWartości statystyk z prób boostrapowych:\n[-0.18332849], [-1.21537352], [1.64628473], [1.06552535], [-0.71420173], ... (i 95 pozostałych)\n\n\n"
]
},
{
"output_type": "display_data",
"data": {
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\n"
},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"p, t, ts = bootstrap_dependent(weights_before, weights_after)\n",
"ts = [x[0] for x in ts]\n",
"draw_distribution(ts, t)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Wniosek\n",
"\n",
"???"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false,
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": []
}
],
"metadata": {
"interpreter": {
"hash": "1b132c2ed43285dcf39f6d01712959169a14a721cf314fe69015adab49bb1fd1"
},
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.10-final"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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