{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "arctic-probe",
   "metadata": {
    "hide_input": true,
    "toc": true
   },
   "source": [
    "<h1>Inhalt<span class=\"tocSkip\"/>\n",
    "</h1>\n",
    "<div class=\"toc\">\n",
    "\t<ul class=\"toc-item\">\n",
    "\t\t<li>\n",
    "\t\t\t<span>\n",
    "\t\t\t\t<a href=\"#Inferenzstatistische-Verfahren\" data-toc-modified-id=\"Inferenzstatistische-Verfahren-1\">Inferenzstatistische Verfahren</a>\n",
    "\t\t\t</span>\n",
    "\t\t\t<ul class=\"toc-item\">\n",
    "\t\t\t\t<li>\n",
    "\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t<a href=\"#Prüfen-von-Voraussetzungen\" data-toc-modified-id=\"Prüfen-von-Voraussetzungen-1.1\">Prüfen von Voraussetzungen</a>\n",
    "\t\t\t\t\t</span>\n",
    "\t\t\t\t\t<ul class=\"toc-item\">\n",
    "\t\t\t\t\t\t<li>\n",
    "\t\t\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t\t\t<a href=\"#Normalverteilung-der-Daten\" data-toc-modified-id=\"Normalverteilung-der-Daten-1.1.1\">Normalverteilung der Daten</a>\n",
    "\t\t\t\t\t\t\t</span>\n",
    "\t\t\t\t\t\t</li>\n",
    "\t\t\t\t\t\t<li>\n",
    "\t\t\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t\t\t<a href=\"#Varianzhomogenität\" data-toc-modified-id=\"Varianzhomogenität-1.1.2\">Varianzhomogenität</a>\n",
    "\t\t\t\t\t\t\t</span>\n",
    "\t\t\t\t\t\t</li>\n",
    "\t\t\t\t\t\t<li>\n",
    "\t\t\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t\t\t<a href=\"#Intervallskalierte-Daten\" data-toc-modified-id=\"Intervallskalierte-Daten-1.1.3\">Intervallskalierte Daten</a>\n",
    "\t\t\t\t\t\t\t</span>\n",
    "\t\t\t\t\t\t</li>\n",
    "\t\t\t\t\t\t<li>\n",
    "\t\t\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t\t\t<a href=\"#Unabhängigkeit\" data-toc-modified-id=\"Unabhängigkeit-1.1.2\">Unabhängigkeit</a>\n",
    "\t\t\t\t\t\t\t</span>\n",
    "\t\t\t\t\t\t</li>\n",
    "\t\t\t\t\t</ul>\n",
    "\t\t\t\t</li>\n",
    "\t\t\t\t<li>\n",
    "\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t<a href=\"#Korrelationsanalyse\" data-toc-modified-id=\"Korrelationsanalyse-1.2\">Korrelationsanalyse</a>\n",
    "\t\t\t\t\t</span>\n",
    "\t\t\t\t</li>\n",
    "\t\t\t\t<li>\n",
    "\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t<a href=\"#Regressionsanalyse\" data-toc-modified-id=\"Regressionsanalyse-1.3\">Regressionsanalyse</a>\n",
    "\t\t\t\t\t</span>\n",
    "\t\t\t\t</li>\n",
    "\t\t\t\t<li>\n",
    "\t\t\t\t\t<span>\n",
    "\t\t\t\t\t\t<a href=\"#Mittelwertvergleiche\" data-toc-modified-id=\"Mittelwertvergleiche-1.4\">Mittelwertvergleiche</a>\n",
    "\t\t\t\t\t</span>\n",
    "\t\t\t\t</li>\n",
    "\t\t\t</ul>\n",
    "\t\t</li>\n",
    "\t</ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "logical-underground",
   "metadata": {},
   "source": [
    "# Inferenzstatistik\n",
    "\n",
    "Bisher haben Sie bereits einige Verfahren zur deskriptiven Analyse von Daten kennengelernt. Die Rückschlüsse und Aussagen, die Sie aus diesen Analysen ableiten können, lassen sich jedoch nicht verallgemeinern und nur auf den analysierten Datensatz bzw. die analysierte Stichprobe beziehen. Je nach Zielsetzung der Untersuchung reicht dies jedoch nicht aus und es wird angestrebt, die Erkenntnisse auf die Grundgesamtheit zu generalisieren. Verfahren und Methoden, mit denen dies unter bestimmten Voraussetzungen möglich ist, lernen Sie nun in diesem Jupyter Notebook kennen. Bevor Sie an dieser Stelle jedoch weiterarbeiten, lesen Sie bitte die Kapitel 1 und 2 aus Field et al. (2012) sowie Abschnitt 12.5.1 aus Döring und Bortz (2016). \n",
    "\n",
    "Nachdem Sie die angegebenen Kapitel gelesen haben, sollten Sie nun die Begriffe Hypothese und Theorie kennen und wissen was eine Zusammenhangs-, eine Unterschieds- oder eine Veränderungshypothese ist. Auch der Unterschied zwischen abhängigen und unabhängigen Variablen ist Ihnen jetzt bekannt und Sie wissen, was ein statistisches Modell ist und können mit den Begriffen einseitiger und zweiseitiger Signifikanztest, Nullhypothese und Alternativhypothese sowie $\\alpha$- und $\\beta$-Fehler, Effektgröße und Teststärke (*power*) etwas anfangen. Sie wissen um verschiedene Häufigkeitsverteilungen und kennen die besondere Bedeutung der Standardnormalverteilung sowie die Möglichkeit mittels z-Transformation jeder normalverteilte Variable in eine Standardnormalverteilung zu überführen.\n",
    "\n",
    "Im Folgenden wollen wir uns mit einigen inferenzstatistischen Testverfahren, deren Voraussetzungen und wie diese zu prüfen sind, näher befassen.\n",
    "\n",
    "## Inferenzstatistische Verfahren\n",
    "\n",
    "### Prüfen von Voraussetzungen\n",
    "Die statistischen Tests, die Sie in diesem Jupyter Notebook kennenlernen werden, gehören zu der Gruppe der so genannten **parametrischen Tests**. Diese Tests nehmen folgende vier Voraussetzungen als gegeben an:\n",
    "\n",
    "- Normalverteilung der Daten (Achtung: unterschiedliche Bedeutung bei unterschiedlichen Tests)\n",
    "- Varianzhomogenität zwischen Gruppen\n",
    "- Intervallskalierte Daten\n",
    "- Unabhängigkeit (Achtung: unterschiedliche Bedeutung bei unterschiedlichen Tests)\n",
    "\n",
    "Vor der Durchführung eines parametrischen Tests sind entsprechend die Voraussetzungen auf ihre Erfüllung hin zu testen. Stellt sich bei der Überprüfung der Voraussetzungen heraus, dass eine oder mehrere nicht erfüllt sind, heißt dies nicht, dass Sie die Daten entsorgen müssen und eine Untersuchung nicht möglich ist. Sie müssen an dieser Stelle auf einen sogenannten nicht-parametrischen Test ausweichen. Diese sind in der Berechnung häufig komplizierter oder die Berechnung dauert unter Umständen länger. Auch eine Transformation der Daten stellt eine Möglichkeit dar. Lassen Sie sich in diesen Fällen im Zweifel noch einmal beraten.\n",
    "\n",
    "#### Normalverteilung der Daten\n",
    "\n",
    "Wie oben bereits angedeutet, hat die Voraussetzung hinsichtlich der Normalverteilung der Daten unterschiedliche Bedeutungen im Zusammenhang verschiedener statistischer Tests. \n",
    "\n",
    "In vielen statistischen Tests (z.B. t-Test) bezieht sich die Voraussetzung zur Normalverteilung auf die Stichprobenverteilung[<sup>1</sup>](#footnote1 \"Erläuterung Stichprobenverteilung\") und nicht -- wie irrtümlich schnell angenommen -- auf die Normalverteilung der Daten in der Stichprobe selbst. Die Stichprobenverteilung ist jedoch i.d.R. unbekannt oder lässt sich nur mit großem Aufwand ermitteln. Jetzt wissen Sie bereits aus der Literatur, dass das **zentrale Grenzwerttheorem** besagt, dass die Stichprobenverteilung für ausreichend große Stichproben[<sup>2</sup>](#footnote2 \"In der Literatur wird häufig ein Wert von mind. 30 angegeben.\") in eine Normalverteilung übergeht (vgl. z.B. Field et al. 2012, S. 43 oder Bortz und Schuster 2010, S. 86). Sind die Stichprobendaten normalverteilt, so kann auf Basis des zentralen Grenzwerttheorems auch angenommen werden, dass die Stichprobenverteilung einer Normalverteilung folgt. Dies heißt im Umkehrschluss aber nicht, dass die Stichprobenverteilung es nicht trotzdem sein kann, wenn die Daten der Stichprobe selbst nicht normalverteilt sind. \n",
    "\n",
    "Im Kontext von **Allgemeinen lineare Modellen (ALM)** (engl. *general linear models (GLM)*) zu denen z.B. die Regression gehört, wird als Voraussetzung angenommen, dass die Fehler eines Modells normalverteilt sind. Dies werden wir an entsprechender Stelle noch vertiefen.\n",
    "\n",
    "Neben visuellen Möglichkeiten zur Überprüfung auf Normalverteilung und der Betrachtung der Werte für die Schiefe und Wölbung der Häufigkeitsverteilung, gibt es auch entsprechende statistische Tests. Meist ist es ratsam alle Möglichkeiten auszuschöpfen, um eine qualifizierte Entscheidung hinsichtlich der Normalverteilung treffen zu können, da auch der weiter unten besprochene **Shapiro-Wilk-Test** Schwächen aufweist.\n",
    "\n",
    "##### Visuelle Prüfung auf Normalverteilung <a class=\"tocSkip\">\n",
    "Eine Möglichkeit Daten visuell auf Normalverteilung zu prüfen, stellen Histogramme dar. Diese haben Sie bereits im dritten Jupyter Notebook kennengelernt. Um die Einschätzung nicht vollständig der Subjektivität des Betrachters zu überlassen, ist es empfehlenswert die Kurve einer Normalverteilung darüber zu legen.\n",
    "\n",
    "Wir lesen im Folgenden den Datensatz `sample_data_final.RDS` ein, hier sind einige Bereinigungen bereits enthalten und die Zahlen für Größe und Gewicht für nachfolgende Demonstrationszwecke noch einmal etwas angepasst."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "abroad-continent",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Beispieldaten einlesen\n",
    "sample_data <- readRDS(\"data/sample_data_final.RDS\")\n",
    "str(sample_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "integral-diabetes",
   "metadata": {},
   "outputs": [],
   "source": [
    "library(ggplot2) #Graphikbibliothek laden\n",
    "library(repr)\n",
    "options(repr.plot.width=5, repr.plot.height=4) #Größe der Abbildung festlegen\n",
    "\n",
    "ggplot(sample_data, aes(Alter)) + \n",
    "    #Mit aes(y=..density..) erreichen wir, dass nicht die absoluten Häufigkeiten, sondern die Dichtefunktion gezeichnet wird.\n",
    "    #Diese Option benötigen wir, um die Normalverteilungskurve darüber zu legen.\n",
    "    geom_histogram(aes(y=..density..), bins = nclass.Sturges(sample_data$Alter), \n",
    "                   color=\"darkblue\", fill=\"cornflowerblue\") + #Balken farblich gestalten (schlechte Sichtbarkeit der Kurve bei default-Farben)\n",
    "    #Diese Funktion errechnet und zeichnet eine Normalverteilungskurve mit Mittelwert und Standardabweichung der Altersdaten.\n",
    "    stat_function(fun=dnorm, args=list(mean=mean(sample_data$Alter, na.rm=TRUE), sd = sd(sample_data$Alter, na.rm=TRUE)), \n",
    "                  color=\"darkred\", size=1) #Linie einfärben und etwas dicker darstellen\n"
   ]
  },
  {
   "attachments": {
    "Histogramm_Normalverteilungspr%C3%BCfung.png": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "rural-journey",
   "metadata": {
    "hide_input": false
   },
   "source": [
    "Wenn Sie die überabarbeiteten Daten verwendet haben, sollte Ihre Graphik wie folgt aussehen:![Histogramm_Normalverteilungspr%C3%BCfung.png](attachment:Histogramm_Normalverteilungspr%C3%BCfung.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adaptive-assignment",
   "metadata": {},
   "source": [
    "Eine weitere Möglichkeit die Daten visuell auf eine Normalverteilung zu prüfen, stellen sogenannte Quantil-Quantil-Diagramme oder kurz Q-Q-Diagramme dar. Diese ermöglichen die Verteilungen zweier Variablen zu vergleichen, in unserem Fall vergleichen wir die beobachteten Werte mit den theoretischen Werten aus der Normalverteilung mit gleichem Mittelwert und gleicher Standardabweichung. Liegt eine Normalverteilung vor, sollten die abgetragenen Punkte eine Gerade bilden. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "timely-boulder",
   "metadata": {
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "ggplot(sample_data, aes(sample=Alter)) + stat_qq() +\n",
    "    stat_qq_line(color=\"darkred\", size=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "potential-charlotte",
   "metadata": {
    "hide_input": false
   },
   "source": [
    "Die visuelle Prüfung zeigt nur geringe Abweichungen von der Normalverteilung an den beiden Enden der Verteilung.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "buried-prior",
   "metadata": {},
   "source": [
    "##### Schiefe und Wölbung der Verteilung <a class=\"tocSkip\">\n",
    "Die Schiefe und Wölbung einer Verteilung sind uns bereits im dritten Jupyter Notebook begegnet. Wir hatten dort bereits angesprochen, dass bei einer Normalverteilung diese Werte `0` betragen. Empirische Daten werden i.d.R. nicht der idealen Normalverteilungskurve folgen, daher werden Sie weder für die Schiefe noch für die Wölbung exakt diesen Wert erhalten. Die Abweichungen sollten jedoch nur gering sein. \n",
    "    \n",
    "Die Beispieldaten weisen mit Werten von $-0,06$ für die Schiefe und $0,21$ für die Wölbung in beiden Fällen recht geringe Werte auf. Auch diese Prüfung hinsichtlich der Normalverteilung kann daher als \"bestanden\" angesehen werden. Weichen die Werte für die Schiefe oder die Wölbung deutlich von 0 ab, so stellt die Transformation von Daten häufig ein probates Mittel dar. Eine gute Darstellung zu gängigen <span id=\"Transformationsverfahren\">Transformationsverfahren</span> finden Sie z.B. bei Field et al. (2012, S. 191 ff.)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "champion-object",
   "metadata": {},
   "outputs": [],
   "source": [
    "library(psych)\n",
    "describe(sample_data$Alter)[c(\"skew\", \"kurtosis\")]"
   ]
  },
  {
   "attachments": {
    "Shapiro-Wilk-Test.PNG": {
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    "##### Shapiro-Wilk-Test<a class=\"tocSkip\">\n",
    "    \n",
    "Der Shapiro-Wilk-Test ist ein statistischer Test, der untersucht, ob die Verteilung einer Variablen von einer vergleichbaren Normalverteilung basierend auf Mittelwert und Standardabweichung abweicht oder nicht. Ähnlich sind wir auch bereits bei der graphischen Prüfung vorgegangen.\n",
    "    \n",
    "Die $H_0$-Hypothese des Shapiro-Wilk-Tests lautet: *Die Daten der Stichprobe sind normalverteilt.* Diese Hypothese möchten wir nicht ablehnen, der Test gilt entsprechend als \"bestanden\", wenn der Shapiro-Wilk-Test kein signifikantes Ergebnis liefert. In diesem Fall können wir die $H_0$-Hypothese beibehalten.\n",
    "    \n",
    "Zu beachten gilt es darüber hinaus, dass der Test für große Stichproben (i.d.R. > 200) schnell ein signifikantes Ergebnis liefert, auch wenn die Abweichungen von der Normalverteilung nur gering sind. Sie sollten daher zusätzlich immer mindestens die graphische Darstellung der Daten bei der Überprüfung mitbetrachten (vgl. z.B. Field et al. 2012, S. 182).\n",
    "    \n",
    "Für die Durchführung des Tests in `R` gibt es wie üblich verschiedene Möglichkeiten. Eine stellt die folgende Funktion dar:\n",
    "```R\n",
    "shapiro.test(zu_testende_variable)\n",
    "```\n",
    "Diese liefert im Ergebnis zwei Werte: die Teststatistik (*W*) und den Signifikanzwert (*p-value*). Liegt letzterer über dem gewählten Signifikanzniveau von $\\alpha = 0,05$, $0,01$ oder $0,001$, so kann die Nullhypothese nicht abgelehnt werden und wir können vorbehaltlich der Irrtumswahrscheinlichkeit davon ausgehen, dass die Daten der Stichprobe für diese Variable normalverteilt sind. Für große Stichproben sollte dabei ein kleineres Signifikanzniveau gewählt werden.\n",
    "    \n",
    "Für unsere Altersvariable erhalten wir folgendes Ergebnis:\n",
    "    ![Shapiro-Wilk-Test.PNG](attachment:Shapiro-Wilk-Test.PNG)\n",
    "Der Test ist auf keinem der oben genannten Signifikanzniveaus signifikant (W = 0,997, p = 0,134), somit kann eine Normalverteilung der Stichprobendaten für diese Variable angenommen werden und schlussendlich auch für die Stichprobenverteilung. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "determined-portrait",
   "metadata": {},
   "outputs": [],
   "source": [
    "shapiro.test(sample_data$Alter)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "alert-elevation",
   "metadata": {},
   "source": [
    "##### Aufgabe<a class=\"tocSkip\">\n",
    "Führen Sie eine Prüfung auf Normalverteilung für die Variablen Größe und Gewicht durch."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "tested-messenger",
   "metadata": {},
   "outputs": [],
   "source": []
  },
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   "cell_type": "code",
   "execution_count": null,
   "id": "recent-performer",
   "metadata": {},
   "outputs": [],
   "source": []
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    "Bei der Bearbeitung der Aufgabe sollte Ihnen mindestens bei der Variable Gewicht eine deutliche Abweichung von der Normalverteilung aufgefallen sein. Das Histogramm sollte Ihnen eine bimodale Verteilung gezeigt haben und der Q-Q-Plot sollte entfernt an ein auf der Seite liegendes *S* erinnern. Darüber hinaus ist der Shapiro-Wilk-Test für diese Variable hochsignifikant (p-value < 2.2e-16 -> p-value < 0,00000000000000022).\n",
    "\n",
    "Häufig sind solche Verteilungsformen durch verschiedene Gruppen in den Daten verursacht. Betrachten wir das Histogramm einmal getrennt nach Geschlechtern, erhalten wir folgende Darstellungen:"
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     ]
    }
   },
   "cell_type": "markdown",
   "id": "4665b5ed",
   "metadata": {},
   "source": [
    "![Normalverteilung_Gruppen-2.svg](attachment:Normalverteilung_Gruppen-2.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c50fa1f",
   "metadata": {},
   "source": [
    "Eine solche getrennte Betrachtung benötigen Sie beispielsweise, wenn Sie entsprechende Gruppenunterschiede mit einem statistischen Test untersuchen wollen. Wir kommen darauf an gegebener Stelle noch einmal zurück.\n",
    "\n",
    "In nachfolgenden Zellen finden Sie den Code zur Erzeugung der entsprechenden Graphiken. Auch der Shapiro-Wilk-Test sowie die Kennzahlen zur Schiefe und Wölbung werden erzeugt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "482ba07b",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Separate Datensätze für die einzelnen Geschlechter anlegen.\n",
    "sample_data_w <- sample_data[sample_data$Geschlecht==\"w\",]\n",
    "sample_data_m <- sample_data[sample_data$Geschlecht==\"m\",]\n",
    "sample_data_d <- sample_data[sample_data$Geschlecht==\"d\",]\n",
    "\n",
    "#Überprüfung der Normalverteilung für Teilgruppe der weiblichen Probanden\n",
    "ggplot(sample_data_w, aes(Gewicht)) + \n",
    "    geom_histogram(aes(y=..density..), bins = nclass.Sturges(sample_data_w$Gewicht), color=\"darkblue\", fill=\"cornflowerblue\") + \n",
    "    stat_function(fun=dnorm, args=list( mean=mean(sample_data_w$Gewicht, na.rm=TRUE), sd = sd(sample_data_w$Gewicht, na.rm=TRUE)), color=\"darkred\", size=1)\n",
    "ggplot(sample_data_w, aes(sample=Gewicht)) + stat_qq() + stat_qq_line(color=\"darkred\", size=1)\n",
    "describe(sample_data_w$Gewicht)[c(\"skew\", \"kurtosis\")]\n",
    "shapiro.test(sample_data_w$Gewicht)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f1349213",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Überprüfung der Normalverteilung für Teilgruppe der männlichen Probanden\n",
    "ggplot(sample_data_m, aes(Gewicht)) + \n",
    "    geom_histogram(aes(y=..density..), bins = nclass.Sturges(sample_data_m$Gewicht), color=\"darkblue\", fill=\"cornflowerblue\") + \n",
    "    stat_function(fun=dnorm, args=list( mean=mean(sample_data_m$Gewicht, na.rm=TRUE), sd = sd(sample_data_m$Gewicht, na.rm=TRUE)), color=\"darkred\", size=1)\n",
    "ggplot(sample_data_m, aes(sample=Gewicht)) + stat_qq() + stat_qq_line(color=\"darkred\", size=1)\n",
    "describe(sample_data_m$Gewicht)[c(\"skew\", \"kurtosis\")]\n",
    "shapiro.test(sample_data_m$Gewicht)"
   ]
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    "#Überprüfung der Normalverteilung für Teilgruppe der diversen Probanden\n",
    "#Achtung: Aufgrund der geringeren Fallzahl fallen einzelne Fälle hier stark ins Gewicht (siehe z.B. kurtosis)\n",
    "ggplot(sample_data_d, aes(Gewicht)) + \n",
    "    geom_histogram(aes(y=..density..), bins = nclass.Sturges(sample_data_d$Gewicht), color=\"darkblue\", fill=\"cornflowerblue\") + \n",
    "    stat_function(fun=dnorm, args=list( mean=mean(sample_data_d$Gewicht, na.rm=TRUE), sd = sd(sample_data_d$Gewicht, na.rm=TRUE)), color=\"darkred\", size=1)\n",
    "ggplot(sample_data_d, aes(sample=Gewicht)) + stat_qq() + stat_qq_line(color=\"darkred\", size=1)\n",
    "describe(sample_data_d$Gewicht)[c(\"skew\", \"kurtosis\")]\n",
    "shapiro.test(sample_data_d$Gewicht)"
   ]
  },
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    "#Anmerkung: Es ist auch möglich mit Hilfe der Funktion by() die Kennzahlen direkt nach Gruppen getrennt zu errechnen.\n",
    "#INDICES erhält die gruppentrennende Variable\n",
    "#FUN die Funktion, die berechnet werden soll\n",
    "by(sample_data$Gewicht, INDICES = sample_data$Geschlecht, FUN = describe)\n",
    "by(sample_data$Gewicht, INDICES = sample_data$Geschlecht, FUN = shapiro.test)"
   ]
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    "#### Varianzhomogenität\n",
    "Wie oben bereits angesprochen, existieren weitere Voraussetzungen, die häufig im Kontext der Anwendung parametrischer Tests erfüllt sein müssen. Dazu zählt auch die Varianzhomogenität, unter der die Varianz der Zielvariablen auf jeder Stufe Einflussvariable(n) gleich sein sollte. In nachfolgender Abbildung sehen Sie eine beispielhafte Darstellung für die Varianz der Variablen Alter für jede Merkmalsausprägung der Variablen Bildungsabschluss."
   ]
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   "source": [
    "options(repr.plot.width=12, repr.plot.height=8) \n",
    "ggplot(sample_data, aes(Bildungsabschluss, Alter)) + geom_point()"
   ]
  },
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   "source": [
    "##### Levene-Test <a class=\"tocSkip\">\n",
    "Da eine graphische Prüfung allein durch subjektive Deutung häufig sowohl in die eine als auch in die andere Richtung interpretiert werden kann. Soll auch an dieser Stelle ein statistischer Test vorgestellt werden, mit dessen Hilfe eine Prüfung auf Varianzhomogenität vollzogen werden kann. \n",
    "\n",
    "Die Nullhypothese des Levene-Tests lautet: Die Varianzen der getesteten Variable sind über alle Stufen der gruppierenden Variable gleich. \n",
    "    \n",
    "Ist das Ergebnis des Levene-Tests signifikant (p-value $\\leq$ 0.05), dann muss die Nullhypothese abgelehnt werden und es ist davon auszugehen, dass die Voraussetzung der Varianzhomogenität nicht gegeben ist. \n",
    "    \n",
    "Eine Implementierung des Levene-Tests finden Sie mit der Funktion `leveneTest()` in der Bibliothek `car`. Diese Bibliothek ist nicht in der Basisinstallation von Anaconda enthalten und muss daher nachträglich installiert werden.\n",
    "```R\n",
    "leveneTest(Zielvariable, gruppenbildende Variable)\n",
    "```"
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    "#install.packages(\"car\")\n",
    "library(car)\n",
    "#Da die Funktion leveneTest() einen DataFrame zurückliefert, setzen wir die Funktion in eine print-Anweisung,\n",
    "#um der Darstellungsform im Jupyter Notebook auszuweichen, die hier leider einige Information unterdrückt.\n",
    "print(leveneTest(sample_data$Alter, group = sample_data$Bildungsabschluss))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bed7dc83",
   "metadata": {},
   "source": [
    "Im vorliegenden Fall liefert der Test kein signifikantes Ergebnis, es kann daher davon ausgegangen werden, dass Varianzhomogenität gegeben ist. Die Varianzen der Variable Alter sind für die verschiedenen Bildungsabschlüsse homogen. Geben Sie als Ergebnis neben der Teststatistik die beiden verschiedenen <span id=\"Freiheitsgrade\">Freiheitsgrade</span>[<sup>3</sup>](#footnote3 \"Anzahl frei variierbarer Abweichungen (vgl. Döring und Bortz 2016, S. 647)\") sowie den Signifikanzwert, der sich in der Ausgabe unter `Pr(>F)` findet, an: F(6,979) = 0,102, p = 0,996. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f8a67df",
   "metadata": {},
   "source": [
    "#### Intervallskalierte Daten\n",
    "Die Prüfung der Intervallskalierung erfolgt nicht mit einem statistischen Test oder graphisch, sondern anhand logischer Überlegungen. Lesen Sie dazu ggf. noch einmal Kapitel 1 bei Field et al. (2012) oder Abschnitt 8.4 bei Döring und Bortz (2016).\n",
    "\n",
    "#### Unabhängigkeit\n",
    "Die Voraussetzung der Unabhängigkeit variiert in ihrer Art in Abhängigkeit des gewählten statistischen Tests. Die Unabhängigkeit kann z.B. für die Teilnehmer (keine Geschwister oder Ehepaare) vorausgesetzt werden oder für die verschiedenen Messungen bei verschiedenen Probanden. Unabhängigkeit kann sich aber auch auf Fehler in einem Modell beziehen (z.B. bei Regressionen). Die verschiedenen Varianten werden bei den jeweiligen statistischen Tests noch einmal genauer erläutert."
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         id="tspan853">        Pearson's product-moment correlation</tspan><tspan
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    "### Korrelationsanalyse\n",
    "\n",
    "Den Begriff Korrelation haben Sie bereits im Jupyter Notebook zur bivariaten deskriptiven Datenanalyse kennengelernt und wir haben uns detailliert mit der Herleitung der Formel für die Berechnung des Korrelationskoeffizienten nach Pearson befasst. Bisher konnten Sie damit jedoch lediglich Aussagen über den Zusammenhang zweier Variablen innerhalb Ihrer Stichprobe treffen. \n",
    "\n",
    "In diesem Abschnitt betrachten wir nun, wie mit Hilfe eines statistischen Tests ausgehend von den Daten in der Stichprobe Rückschlüsse auf die Population gezogen werden können. Die ungerichtete Nullhypothese lautet in diesem Zusammenhang: \"*Es besteht kein linearer Zusammenhang zwischen den beiden Variablen X und Y*\". Sie erinnern sich sicher, dass bei einem Korrelationskoeffizient von Null davon ausgegangen wird, dass kein linearer Zusammenhang zwischen den Variablen besteht. Formal können wir die Nullhypothese daher wie folgt notieren: $H_0: r = 0$. Die Alternativhypothese laut in diesem Fall: \"*Es besteht ein linearer Zusammenhang zwischen den beiden Variablen X und Y ($H_1: r \\neq 0$)*\"\n",
    "\n",
    "Die Funktion `cor()`, die wir bisher zur Berechnung von Korrelationen herangezogen haben, ist nicht in der Lage, einen statistischen Test durchzuführen. Dies lässt sich aber alternativ mit der Funktion `cor.test()` erreichen, die neben dem Signifikanztest auch das Konfidenzintervall[<sup>4</sup>](#footnote4 \"Intervall, innerhalb dessen mit einer gegebenen Wahrscheinlichkeit (i.d.R. 95%) der wahre Wert der Grundgesamtheit\") berechnet.\n",
    "\n",
    "```R\n",
    "cor.test(sample_data$Gewicht, sample_data$Groesse)\n",
    "```\n",
    "Für unseren Beispieldatensatz erhalten wir folgendes Ergebnis für die Variablen Größe und Gewicht.\n",
    "\n",
    "![Pearson_Korrelation_Test.svg](attachment:Pearson_Korrelation_Test.svg)"
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    "In der Zeile mit Ziffer 1 sehen Sie neben der Teststatistik (t = 49,4) und den Freiheitsgraden (df = 998) die ermittelte Irrtumswahrscheinlichkeit (p < 0,001). Das Ergebnis ist hochsignifikant, die Nullhypothese kann daher abgelehnt werden. Sie finden unter Ziffer 2 zusätzlich die Angabe zur Alternativhypothese. Da die Funktion auch das Testen gerichteter Hypothesen unterstützt, können Sie an dieser Stelle noch einmal überprüfen, ob Sie die richtigen Einstellungen vorgenommen haben. Die Angabe zum errechneten Korrelationskoeffizienten finden Sie unter Ziffer 4. Dies ist der Wert, den Sie auch bei der Berechnung mit der Funktion `cor()` erhalten. Die Angabe zum Konfidenzintervall finden Sie bei Ziffer 3, hier ist auch die die Wahrscheinlichkeit angegeben, mit der der Korrelationskoeffizient der Population innerhalb des Intervalls liegt. Das Intervall ist hier mit einer Spanne von 0,60 bis 0,67 recht klein, wichtig ist in jedem Fall, dass es die Null nicht einschließt. Wäre dies der Fall, könnte nicht davon ausgegangen werden, dass in der Population ein positiver Zusammenhang zwischen den untersuchten Variablen Größe und Gewicht besteht."
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    "cor.test(sample_data$Gewicht, sample_data$Groesse) #ungerichtete Hypothese mit Standardeinstellungen testen"
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"
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       x="0"
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    "Haben Sie durch vorangegangene Studien oder aus der Literatur bereits eine starke Vermutung über die Richtung des Zusammenhangs (positive oder negative Korrelation), dann sollten Sie eine gerichtete Hypothese formulieren. In diesem Fall wird dann ein einseitiger Signifikanztest durchgeführt. Zur Erinnerung an das Wissen aus der Lektüre: Da die Richtung des zu prüfenden Zusammenhangs bereits vorgegeben ist, muss die Irrtumswahrscheinlichkeit nicht für eine Prüfung auf beiden Seiten aufgeteilt werden. Der Bereich, in dem die Nullhypothese abgelehnt werden kann, ist daher entsprechend größer.\n",
    "\n",
    "![Einseitiger_Zweiseitiger_Signifikanztest.svg](attachment:Einseitiger_Zweiseitiger_Signifikanztest.svg)\n",
    "(Graphische Darstellung in Anlehnung an Dörung und Bortz 2016, S. 668)"
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    "Für den Zusammenhang von Größe und Gewicht können wir folgende gerichtete Hypothese formulieren: \"*Zwischen Größe und Gewicht besteht ein positiver Zusammenhang.*\" (formal: $H_1: r > 0$). Die Nullhypothese lautet entsprechend \"*Zwischen Größe und Gewicht besteht kein Zusammenhang oder ein negativer Zusammenhang*\" (formal: $H_0: r \\leq 0$).\n",
    "\n",
    "```R\n",
    "#Spezifizieren der Hypothesenrichtung über den Parameter alternative, der die Werte \n",
    "#\"two.sided\" (default), \"less\" und \"greater\" annehmen kann\n",
    "cor.test(sample_data$Gewicht, sample_data$Groesse, alternative = \"greater\")\n",
    "```\n",
    "\n",
    "Da der Wert für p bereits beim zweiseitigen Test die kleinste von `R` ausgegebene Zahl unterschritten hat, ähneln sich die Ausgaben der beiden Tests natürlich."
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    "Erhalten wir bei einem statistischen Test ein signifikantes Ergebnis, ist im zweiten Schritt die Effektgröße zu bestimmen. Ein signifikantes Ergebnis bedeutet nicht automatisch, dass es sich auch um einen Effekt von Bedeutung handelt. Dies ist insbesondere bei großen Stichproben wichtig, da mit diesen auch kleine Effekte nachweisbar sind.\n",
    "\n",
    "Um die Effektgröße über verschiedene Untersuchungen hinweg vergleichbar zu machen, sollte ein standardisiertes Maß verwendet werden. Eines der gängigen standardisierten Maße für Zusammenhangshypothesen stellt der Korrelationskoeffizient $r$ selbst dar. Sie erinnern sich vielleicht, dass wir dies bereits im Jupyter Notebook zur bivariaten deskriptiven Datenanalyse angesprochen haben. Folgende Einschätzungen zur Größe des Effekts hatten wir in diesem Zusammenhang festgehalten (mit dem Hinweis, dass diese nur als Daumenregel zu betrachten sind und eine konkrete Einschätzung immer im Kontext der Forschungsliteratur erfolgen sollte):\n",
    "- großer Effekt: $r \\geq \\pm 0,5$\n",
    "- mittelstarker Effekt: $r \\geq\\pm 0,3$\n",
    "- kleiner Effekt: $r \\geq\\pm 0,1$\n",
    "\n",
    "Der quadrierte Korrelationskoeffizient (Determinationskoeffizient $R^2$ gibt dabei den Anteil der Varianzaufklärung durch den Merkmalszusammenhang an. Für unser Beispiel bedeutet dies, dass durch den Zusammenhang zwischen Größe und Gewicht etwa 70,9% der Varianz in den Daten erklärt wird. Die übrigen knapp 30% sind daher auf andere Effekte zurückzuführen."
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    "#Berechnung von R^2, ausgedrückt in Prozentwerten, gerundet auf eine Stelle nach dem Komma\n",
    "round(cor(sample_data$Gewicht, sample_data$Groesse)^2 *100, 1)"
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    "Bei Ihrer Analyse zur Überprüfung auf Normalverteilung haben Sie zuvor festgestellt, dass diese für Größe und Gewicht nur innerhalb der jeweiligen Geschlechtergruppen gegeben war. Die Ergebnisse der obigen Korrelationsanalyse sind daher nicht verlässlich und sie ist entsprechend für die einzelnen Geschlechtergruppen separat durchzuführen. Sie sehen an dieser Stelle, dass die Berechnung auch bei fehlenden Voraussetzungen von `R` durchgeführt wird. Dies gilt auch für andere Statistikprogramme wie `SPSS`. Die Programme nehmen Ihnen zwar die Rechenarbeit ab, denken aber nicht mit. Diese Aufgabe verbleibt bei der bedienenden Person."
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    "##### Aufgabe<a class=\"tocSkip\">\n",
    "Führen Sie die Korrelationsanalyse getrennt für die einzelnen Geschlechter durch. Welche Beobachtung können Sie im Vergleich mit dem Ergebnis der zuvor durchgeführten Korrelationsanalyse mit verletzter Voraussetzung der Normalverteilung machen?"
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    "Mit der vorgestellten Funktion `cor.test()` lassen sich nicht nur Korrelationsanalysen für metrische Variablen durchführen, auch statistische Tests für Spearmans $\\rho$ und Kendalls $\\tau$ sind damit möglich. Die Einstellung erfolgt wie bei der Funktion `cor()` über einen Parameter `method`, der neben der Standardeinstellung `pearson` auch die Werte `spearman` und `kendall` annimmt. Wir werden diesen Test hier nicht weiter vertiefen, der interessierte Leser sei an dieser Stelle auf Field (2012), Abschnitt 6.5.5 und 6.5.6 verwiesen. "
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    "### Regressionsanalyse\n",
    "Bei der Betrachtung von Zusammenhängen zwischen zwei oder mehr Variablen interessiert nicht immer nur die Existenz und Stärke eines Zusammenhangs. Häufig wird auch gewünscht, Vorhersagen für eine Variable (Kriteriumsvariable) auf Basis der Ausprägungen einer oder mehrerer anderer Variablen (Prädiktoren) zu tätigen. Dabei nimmt die Genauigkeit der Vorhersage zu, je stärker der Zusammenhang zwischen den betrachteten Variablen ist. Die Regressionsanalyse wird dabei häufig in Untersuchungen eingesetzt, bei denen ein kausaler Zusammenhang zwischen der Kriteriumsvariable und der bzw. den Prädiktoren besteht. Bei der Betrachtung eines einzelnen Prädiktors zur Vorhersage der Ausprägung einer Kriteriumsvariablen spricht von ein einer bivariaten oder einfachen Regression. Werden mehrere Prädiktoren berücksichtigt, handelt es sich um eine multiple Regression.\n",
    "\n",
    "Wir haben bei der Einführung der Korrelation bereits darüber gesprochen, dass für metrische Variablen häufig ein linearer Zusammenhang angenommen werden kann. Ist ein solcher Zusammenhang vorhanden, lässt sich dieser mit einer Geraden beschreiben. Die grundlegende Form eine Geradengleichung kennen Sie gewiss noch aus dem Mathematikunterricht in der Schule:\n",
    "\n",
    "$$ y = a + bx,  \\quad\\quad\\text{mit }\\quad\n",
    "\\begin{eqnarray}\n",
    "    y&= & \\text{ Kriteriumsvariable}\\\\\n",
    "    a&= & \\text{ y-Achsenabschnitt}\\\\\n",
    "    b&= & \\text{ Steigung}\\\\\n",
    "\\end{eqnarray}$$\n",
    "\n",
    "![Gerade-3.svg](attachment:Gerade-3.svg)"
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    "Ist die Steigung (b) positiv, so steigt die Gerade wie in der obigen Graphik und zwischen den betrachteten Variablen besteht ein positiver Zusammenhang. Ist die Steigung negativ, so fällt die Gerade und es besteht ein negativer Zusammenhang zwischen den beiden Variablen. \n",
    "\n",
    "Im Kontext von Regressionen finden Sie in der Literatur häufig nachfolgende Schreibweisen für Regressionsgeraden. Dabei stehen $b_0$ bzw. $\\beta_0$ für den y-Achsenabschnitt (engl. intercept) und $b_1$ bzw. $\\beta_1$ für die Steigung (engl. slope). Die lateinischen Buchstaben beziehen sich in diesem Zusammenhang i.d.R. auf die Stichprobe und die griechischen Buchstaben stehen für die geschätzen Parameter der Grundgesamtheit:\n",
    "\n",
    "$$ y = b_0 + b_1x $$ \n",
    "$$ y = \\beta_0 + \\beta_1x $$ \n",
    "\n",
    "Nun wissen Sie bereits aus den Überlegungen, die wir im Rahmen der Korrelation angestellt haben, dass Zusammenhänge in empirischen Daten in den seltensten Fällen perfekt sind und die Punkte der gemeinsam beobachteten Merkmalsausprägungen zweier Variablen keine perfekte Gerade bilden, sondern sich in einer (elliptischen) Punktwolke verteilen. Grundsätzlich sind viele verschiedene Geraden denkbar, die als Modell für die Beschreibung der Daten dienen können. \n",
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    "![Regressionsgeraden.svg](attachment:Regressionsgeraden.svg)"
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   "cell_type": "markdown",
   "id": "bf5aaaff",
   "metadata": {},
   "source": [
    "Hier kommt jetzt eine zweite Überlegung zum Tragen, die wir bereits an früher angestellt haben: Jedes statistische Modell ist fehlerbehaftet. Betrachten Sie noch einmal obige Graphik. Jeder Punkt, der nicht genau auf einer Geraden liegt, wird durch das Modell (hier: Gerade) mit einem gewissen Fehler (Abweichung von der Geraden) vorhergesagt. Ziel der Regressionsanalyse ist es nun, diejenige Gerade zu finden, bei der die Vorhersagefehler möglichst gering sind. \n",
    "\n",
    "In nachfolgender Graphik sind die Vorhersagefehler für einen Beispieldatensatz mit geringerem Umfang zwecks besserer Darstellbarkeit eingezeichnet. Im Kontext von Regressionen werden die Abweichungen zwischen den Werten, die das Modell vorhersagt (auf der Linie), und den tatsächlich beobachteten Werten als Residuen bezeichnet.\n",
    "\n",
    "![Regressionsgeraden_Residuen.svg](attachment:Regressionsgeraden_Residuen.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8e3b230",
   "metadata": {},
   "source": [
    "Die Ermittlung der Geraden, für die die Vorhersagefehler am geringsten sind, erfolgt mit Hilfe der Methode der kleinsten Quadrate. Sie erinnern sich sicher, dass wir bei der Besprechung der Varianz zu dem Kniff gegriffen haben, die Abweichungen vom Mittelwert zu quadrieren, da die positiven und negativen Abweichungen sich gegenseitig aufgehoben haben. Dies ist auch hier der Fall, weswegen wieder die quadrierten Abweichungen (hier: Residuen) für die Berechnungen verwendet werden. Wir verzichten an dieser Stelle auf die Herleitung der Formel zur Ermittlung der Regressionsgeraden und überlassen die Berechnung dem Statistikprogramm. Der interessierte Leser sei auf Häder (2015, S. 187ff.) verwiesen."
   ]
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  {
   "cell_type": "markdown",
   "id": "46abe7ee",
   "metadata": {},
   "source": [
    "Die Berechnung des <span id=\"regressionsmodell\">Regressionsmodells</span> erfolgt in `R` mit der Funktion `lm()` (=linear model). Die Kriteriumsvariable und die Prädiktoren werden in einer als *Formel* bezeichnete Form an die Funktion übergeben, die eine Tilde[<sup>5</sup>](#footnote5 \"Die Tilde findet sich auf derselben Taste wie das +-Zeichen.\") als Operator nutzt:\n",
    "\n",
    "```R\n",
    "lm(kriteriumsvariable ~ praediktor, data=data_frame_mit_daten)\n",
    "```\n",
    "\n",
    "Mit dem zusätzlichen Parameter `na.action` können Sie spezifizieren, wie mit fehlenden Werten umgegangen werden soll.\n",
    "\n",
    "Im Folgenden ermitteln wir nun exemplarisch das Regressionsmodell für die weiblichen Probanden. Mit diesem wollen wir das Gewicht auf Basis der Größe vorhersagen. Die Kriteriumsvariable ist daher das `Gewicht` und der Prädiktor die `Groesse`. Da die Daten keine fehlenden Werten enthalten, benötigen wir an dieser Stelle keine entsprechende Angabe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "df9b059f",
   "metadata": {},
   "outputs": [],
   "source": [
    "lm(Gewicht ~ Groesse, data=sample_data[sample_data$Geschlecht==\"w\",])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1c74185",
   "metadata": {},
   "source": [
    "In der Ausgabe sehen Sie nun die berechneten Modellparameter: Der y-Achsenabschnitt (Intercept) wird mit $\\approx$ -17,01 angegeben und die Steigung mit $\\approx$ 0,46. Die Regressionsgerade lautet demgemäß für die weiblichen Probanden: \n",
    "\n",
    "$$ Gewicht_i = -17,01 + 0,46 \\cdot Groesse_i$$\n",
    "\n",
    "Die Steigung kann an dieser Stelle als die Veränderung der Kriteriumsvariable betrachtet werden, die mit einer Veränderung des Prädiktors um eine Einheit (hier: cm) einhergeht. Eine um einen Zentimeter größere Person wird gemäß des vorliegenden Modells ca. ein halbes Kilogramm schwerer sein als eine zweite, entsprechend kleinere Person.\n",
    "\n",
    "Bevor wir uns nun damit befassen, wie gut das Modell die Daten repräsentiert und die Voraussetzungen für die Regression erfüllt sind, betrachten wir zunächst eine graphische Darstellungsmöglichkeit. Sie kennen bereits das Erzeugen von Punktwolken und die Bibliothek `ggplot2` liefert eine komfortable Möglichkeit für das Einzeichnen einer Regressionsgeraden. Die Funktion `geom_smooth` übernimmt die Berechnung und das Einzeichnen der Regressionsgeraden und geht in den Standardeinstellungen davon aus, dass die Variable, die auf der y-Achse abgetragen wird, die Kriteriumsvariable darstellt und die Variable auf der x-Achse den Prädiktor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "54bae0f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "#library(ggplot2) Bibliothek laden (kann erforderlich sein, wenn Sie das JN in mehreren Sessions bearbeiten)\n",
    "ggplot(sample_data[sample_data$Geschlecht==\"w\",], aes(Groesse, Gewicht)) + geom_point() + \n",
    "    #geom_smooth errechnet die Regressionsgerade (method = \"lm\") und zeichnet diese ein\n",
    "    #se = FALSE: Unterdrücken der Darstellung des Standardfehlers\n",
    "    #color=\"blue\": Farbe der Regressionsgeraden\n",
    "    geom_smooth(method = \"lm\", se = FALSE, color=\"blue\") "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "230d9c38",
   "metadata": {},
   "source": [
    "##### Aufgabe<a class=\"tocSkip\">\n",
    "Ermitteln Sie anhand der vorliegenden Daten für die männlichen Probanden ein Regressionsmodell und stellen Sie dies graphisch dar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "33762245",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "f2580871",
   "metadata": {},
   "source": [
    "#### Voraussetzungen prüfen\n",
    "Sie wissen bereits, dass eine Voraussetzung für die Durchführung einer Regressionsanalyse in der Normalverteilung der Modellfehler (Residuen) liegt. Das oben bereits errechnete Modell enthält neben den ausgegebenen Modellparametern noch weitere nützliche Informationen, z.B. die Residuen. Es ist daher empfehlenswert das Modell in einer Variablen abzulegen, um auf weitere Elemente zugreifen zu können."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "364eaf26",
   "metadata": {},
   "outputs": [],
   "source": [
    "regressions_modell_weibliche_probanden <- lm(Gewicht ~ Groesse, data=sample_data[sample_data$Geschlecht==\"w\",])\n",
    "regressions_modell_weibliche_probanden #Modell aufrufen und Ergebnis betrachten\n",
    "#Bei Interesse: Struktur des Modells abrufen (Achtung: Ausgabe ist recht lang)\n",
    "#str(regressions_modell_weibliche_probanden)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "313efdc1",
   "metadata": {},
   "source": [
    "Eine schnelle und einfache Möglichkeit, um eine graphische Prüfung der Normalverteilung durchzuführen, bietet `R` wieder mit der Funktion `plot()` aus der Basisbibliothek. Übergeben Sie das Modell an die Funktion, erhalten Sie eine Reihe von Graphiken, u.a. einen Q-Q-Plot der standardisierten Residuen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c32b63ce",
   "metadata": {},
   "outputs": [],
   "source": [
    "plot(regressions_modell_weibliche_probanden)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05d67532",
   "metadata": {},
   "source": [
    "Auch `ggplot2` bietet die Möglichkeit die Residuen graphisch auf Normalverteilung zu prüfen, die Erzeugung der Graphiken ist aber etwas aufwendiger. Ein Beispiel sehen Sie in nachfolgender Zelle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3ef09c2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Dem DataFrame mit den Daten die Residuen aus dem Modell sowie die prognostizierten Werte (für den späteren Gebrauch) hinzufügen\n",
    "sample_data_w <- cbind(sample_data[sample_data$Geschlecht==\"w\",], \n",
    "                       Residuen=regressions_modell_weibliche_probanden$residuals,\n",
    "                       Prognostizierte_Werte = regressions_modell_weibliche_probanden$fitted.values)\n",
    "\n",
    "#Histogramm erstellen\n",
    "ggplot(sample_data_w, aes(Residuen)) + \n",
    "    geom_histogram(aes(y=..density..), bins = nclass.Sturges(sample_data_w$Residuen), \n",
    "                   color=\"darkblue\", fill=\"cornflowerblue\") + \n",
    "    stat_function(fun=dnorm, args=list(mean=mean(sample_data_w$Residuen, na.rm=TRUE), sd = sd(sample_data_w$Residuen, na.rm=TRUE)), \n",
    "                  color=\"darkred\", size=1) \n",
    "\n",
    "#Q-Q-Plot erstellen\n",
    "ggplot(sample_data_w, aes(sample=Residuen)) + stat_qq() + stat_qq_line(color=\"darkred\", size=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6916839a",
   "metadata": {},
   "source": [
    "Auch eine Betrachtung des Shapiro-Wilk-Tests sowie der Werte für die Schiefe und Wölbung der Residuenverteilung können Sie analog zu dem oben gelernten durchführen. Beide Ergebnisse lassen, ebenso wie die Graphiken, keinen Hinweis auf die Verletzung der Voraussetzung die Residuen seien normalverteilt erkennen. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "03bed014",
   "metadata": {},
   "outputs": [],
   "source": [
    "describe(sample_data_w$Residuen)[c(\"skew\", \"kurtosis\")]\n",
    "shapiro.test(sample_data_w$Residuen)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23ca5079",
   "metadata": {},
   "source": [
    "Überdies muss an dieser Stelle  die Voraussetzung auf Varianzhomogenität überprüft werden. Zu diesem Zweck können wir die Verteilung der vom Modell vorhergesagten Werte mit den Residuen gemeinsam graphisch darstellen. Scheinen die Punkte zufällig verteilt zu sein, so ist dies ein starkes Indiz dafür, dass die Voraussetzung nicht verletzt ist. Die `plot()`-Funktion hat uns oben bereits eine solche Graphik ausgegeben. Es handelt sich dabei um die erste der erzeugten Graphiken.\n",
    "\n",
    "Mit `ggplot2` können Sie die Graphik wie folgt erzeugen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e68da909",
   "metadata": {},
   "outputs": [],
   "source": [
    "ggplot(sample_data_w, aes(Prognostizierte_Werte, Residuen)) + geom_point()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e67cc97e",
   "metadata": {},
   "source": [
    "Zusätzlich sind bei einer Regression noch weitere Voraussetzungen zu prüfen. So muss der Zusammenhang zwischen der unabhängigen und abhängigen Variable linear sein. Dies sollte leicht nachvollziehbar sein, da unser Vorhersagemodell eine Gerade darstellt. Ist der Zusammenhang nicht linear, so ist ein lineares Modell schlecht für die Vorhersage geeignet. Häufig lässt sich etwaigen andersartigen Zusammenhänge durch eine Transformation der Daten begegnen (siehe auch Hinweise in Abschnitt [Schiefe und Wölbung der Verteilung](#Transformationsverfahren)). Die Überprüfung kann z.B. visuell mit einer Betrachtung der Punktwolke erfolgen. Wird mehr als ein Prädiktor im Regressionsmodell betrachtet, kommen noch eine Reihe weiterer Voraussetzungen hinzu (siehe z.B. Field et al. 2012, Abschnitt 7.7.2). Auch diese sind zwingend zu prüfen, wenn die Aussagen über die Daten der Stichprobe hinaus generalisiert werden sollen. Bei nicht-Erfüllung stehen z.B. robuste Regressionsmodelle als Alternativen zur Verfügung.\n",
    "\n",
    "Nachdem die Voraussetzungen geprüft sind und keine Verletzung festgestellt wurde, können wir das Modell auch über die Daten hinaus generalisieren. In diesem Zusammenhang ist wieder zu prüfen, ob die ermittelten Modellparameter signifikant sind und wie groß der Effekt bei vorliegender Signifikanz ist.\n",
    "\n",
    "Das zuvor berechnete Modell enthält dabei bereits die genannten Informationen. Aufrufen können wir diese mit Hilfe der Funktion `summary()`, die Sie auch bereits aus der deskriptiven Analyse kennen. Statt den Daten übergeben wir hier jedoch das Regressionsmodell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "46097343",
   "metadata": {
    "scrolled": true
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   "outputs": [],
   "source": [
    "summary(regressions_modell_weibliche_probanden)"
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   "source": [
    "Sie erhalten folgende Ausgabe:\n",
    "\n",
    "![Regression.svg](attachment:Regression.svg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e337ae3",
   "metadata": {},
   "source": [
    "In den Zeilen mit den Ziffern 1 und 2 sehen Sie neben den Modellparametern, die wir bereits aus der vorangegangenen Ausgabe kennen, auch Angaben zur Signifikanz der jeweiligen Parameter. Dabei wird jeweils anhand einen T-Tests (s.u.) geprüft, ob der Parameter signifikant von 0 abweicht. Im Beispiel ist dies für beide Parameter gegeben. \n",
    "\n",
    "In der mit Ziffer 3 gekennzeichneten Zeile finden Sie Angaben zur Effektstärke des gesamten Modells. Diese ist mit $R^2$ angegeben, welches Sie aus der Korrelationsanalyse bereits kennen. Das Modell erklärt hier 33,9% der Varianz in den Daten. Die verbleibenden 66,1% werden durch andere Faktoren, die nicht im Modell berücksichtigt sind, verursacht.\n",
    "\n",
    "In der letzten Zeile (Ziffer 4) finden wir schließlich noch eine Angabe dazu, ob das Modell als gesamtes eine bessere Vorhersage der Daten liefert als das einfachster aller Modelle, der Mittelwert. Auch dieser Test ist hochsignifikant.\n",
    "\n",
    "Zusätzlich können Sie sich mit der Methode `confint()` die Konfidenzintervalle für die Regressionsparameter ausgeben lassen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6d81b5e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "confint(regressions_modell_weibliche_probanden)"
   ]
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  {
   "cell_type": "markdown",
   "id": "985443ae",
   "metadata": {},
   "source": [
    "##### Aufgabe<a class=\"tocSkip\">\n",
    "Prüfen Sie die Voraussetzungen für die Regression der männlichen Probanden und prüfen Sie, ob das Modell sowie seine Parameter signifikant sind. Welche Aussagen können Sie zur Effektstärke machen?\n",
    "    \n",
    "(Wenn Sie alles korrekt durchgeführt haben, sollten Sie folgende Regressionsgerade erhalten: $ Gewicht_i = 3.8 + 0,44 \\cdot Groesse_i$. Der y-Achsenabschnitt wird im Test nicht signifikant, der Regressionsparameter für die Größe ist hingegen hochsignifikant. Das Modell liefert insgesamt aber eine signifikant bessere Schätzung des Gewichts als der Mittelwert. Wenn Sie sich zusätzlich die Konfidenzintervalle der Modellparameter berechnen lassen, dann sehen Sie, dass dieses für den y-Achsenabschnitt und mit Werten zwischen -5,5 und +13,1 auch die Null einschließt. Die Regressionsgerade kann in der Population daher deutlich an der y-Achse verschoben sein.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5cbf341f",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "e16d9cd8",
   "metadata": {},
   "source": [
    "Sie haben nun die einfache lineare Regression kennengelernt, bei der die Werte der Kriteriumsvariablen lediglich mit Hilfe eines Prädiktors vorhergesagt werden. Häufig erreichen diese einfachen Modelle nicht die gewünschte Genauigkeit bei der Vorhersage der Kriteriumsvariablen und es werden weitere Prädiktoren hinzugezogen. Diese multiplen Regressionsmodelle funktionieren nach demselben Prinzip wie die einfache lineare Regression, erfordern aber die Prüfung einiger weiterer Voraussetzungen. Wir werden die multiple Regression an diese Stelle nicht vertiefen, bei Interesse lohnt sich ein Blick in das Kapitel 7 von Field et al. (2012)."
   ]
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   "id": "21a285a6",
   "metadata": {},
   "source": [
    "### Mittelwertvergleiche (t-Test)\n",
    "Die bisher betrachteten statistischen Modelle haben die Untersuchung von Zusammenhangshypothesen ermöglicht. In diesem Abschnitt wollen wir uns mit der Überprüfung von Unterschieds- und Veränderungshypothesen befassen. Diese kommen gerade in experimentellen Untersuchungen zum Einsatz, mit dem Ziel festzustellen, ob eine (absichtlicht herbeigeführte) Veränderung bei einer oder mehreren unabhängigen Variablen zu einer Veränderung bei einer anhängigen Variablen (Zielvariable) führt. In der Informationswissenschaft finden wir solche Untersuchungen z.B. bei Forschung im Retrieval-Kontext, bei der es häufig um die Frage geht, ob eine neu entwickelte Systemkomponente zu einer signifikanten Verbesserung in der Ergebnisqualität eines Retrieval-Systems führt (Bewirkt die neue Systemkomponente einen signifikanten Unterschied?). Aber auch im Kontext von Nutzerstudien finden wir entsprechende Experimente. \n",
    "\n",
    "Grundsätzlich werden zwei verschiedene Untersuchungsdesigns in diesem Zusammenhang unterschieden, wie sie bereits aus der Lektüre wissen: \n",
    "1. Es werden wiederholte Messungen bei derselben Untersuchungsgruppe vorgenommen (within-subject oder repeated-measures-design)\n",
    "2. Es werden zwei oder mehr Gruppen gebildet, die unterschiedlichen Treatments (Behandlungen/Maßnahmen) ausgesetzt werden (between-subjects oder independend design)\n",
    "\n",
    "Die Art des Untersuchungsdesigns hat dabei Einfluss auf die Wahl des statistischen Tests, der im Rahmen der Auswertung zum Einsatz kommen kann. Wir werden im Folgenden exemplarisch die Analyse von Mittelwertvergleichen (t-Test) näher betrachten. Für beide der oben benannten Untersuchungsdesigns gibt es eine Variante des t-Tests. Wird ein Within-Subject-Design gewählt, so ist der t-Test für abhängige Stichproben zu wählen, erfolgt die Untersuchung mit einem Between-Subject-Design, stellt der t-Test für unabhängige Stichproben die richtige Wahl dar.\n",
    "\n",
    "Die zu wiederlegende Nullhypothese lautet bei beiden Testvarianten \"Es besteht kein Unterschied zwischen den Mittelwerten.\" (formal: $H_0: \\bar{x_1} = \\bar{x_2}$). Die Alternativhypothese lautet entsprechend: \"Es besteht ein Unterschied zwischen den Mittelwerten\" (formal: $H_1: \\bar{x_1} \\neq \\bar{x_2}$). Natürlich sind auch gerichtete Varianten der Hypothesen möglich.\n",
    "\n",
    "Der Grundgedanke hinter diesem Test ist, dass die Mittelwerte (annähernd) gleich sind, wenn die Stichproben (Gruppen) aus der gleichen Population kommen. Dies ist der Fall, wenn das Treatment keinen Effekt hat, also die Nullhypothese nicht abgelehnt werden kann. Sie wissen bereits, dass bei unterschiedlichen Stichproben aus derselben Population durchaus Schwankungen in den Mittelwerten auftreten können. Wie groß die Variabilität zwischen den Stichprobenmittelwerten sein kann, kann näherungsweise mit dem Standardfehler[<sup>1</sup>](#footnote1 \"Erläuterung Stichprobenverteilung und Standardfehler\") geschätzt werden. Bei einem kleinen Standardfehler kann davon ausgegangen werden, dass die meisten Stichproben einen ähnlichen Mittelwert haben. Ist dieser hingegen groß, so können starke Schwankungen in den Mittelwerten verschiedener Stichproben aus derselben Population auftreten. Wenn ein großer Standardfehler vorliegt, dann können Unterschiede in den Mittelwerten zweier Stichproben entsprechend auch auf die Schwankungen zwischen den Stichprobenverteilungen zurückgeführt werden und müssen nicht zwangsweise auf das Treatment zurückführbar sein. Der t-Test berücksichtigt daher den Standardfehler bei der Beurteilung des Mittelwertvergleichs. Für den t-Test mit unabhängigen Stichproben sieht die Formel zur Berechnung der Teststatistik wie folgt aus."
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    "![t-Test_Formel.svg](attachment:t-Test_Formel.svg)\n",
    "\n",
    "#### Voraussetzungen\n",
    "Wie bei allen statistischen Tests sind auch bei den t-Tests gewisse Voraussetzungen an die Durchführung geknüpft. Da wir mit dem Test Mittelwerte vergleichen, muss die abhängige Variable mindestens intervallskaliert sein. Darüber hinaus wird angenommen, dass die Stichprobenverteilung normalverteilt ist. Beim t-Test für abhängige Stichproben bezieht sich die Normalverteilungsvoraussetzung auf die Differenzen zwischen den Werten der Messzeitpunkte. Für den <span id=\"t-test-voraussetzungen\">t-Test</span> unabhängiger Stichproben ist selbstredend die Voraussetzung zu erfüllen, dass die beobachteten Werte der unabhängigen Variable für verschiedene Treatments unabhängig sind[<sup>6</sup>](#footnote5 \"Unabhängige Messwerte bei unabhängigen Stichproben.\"). Darüber hinaus ist theoretisch noch die Voraussetzung der Varianzhomogenität zu erfüllen, `R` berechnet hier jedoch automatisch eine Korrektur (Welch t-Test) wodurch eine Verletzung der Voraussetzung nicht mehr ins Gewicht fällt.\n",
    "\n",
    "Im Folgenden führen wir exemplarisch einen t-Test für unabhängige Stichproben durch, mit dessen Hilfe wir untersuchen wollen, ob das Geschlecht einen systematischen Unterschied bei der Größe beobachten lässt. Die Alternativhypothese lautet in diesem Fall wie folgt: \"Es besteht ein Unterschied zwischen der Größe von Frauen und Männern gemessen am Mittelwert.\" Daraus lässt sich als Nullhypothese die entsprechend negierte Alternativhypothese ableiten als: \"Es besteht kein Unterschied zwischen der Größe von Frauen und Männern gemessen am Mittelwert.\"\n",
    "\n",
    "In nachfolgend erzeugter Abbildung und den errechneten deskriptiven Kennzahlen sehen wir bereits einen entsprechenden Unterschied in den Mittelwerten und auch den zugehörigen Schätzwert für den Standardfehler (se). Ob der beobachetete Mittelwertunterschied jedoch auf die Stichprobenverteilung zurückzuführen ist oder auf unterschiedliche Populationen hindeutet, werden wir anschließend mit einem t-Test für unabhängige Stichproben ermitteln (die Gruppe der männlichen und weiblichen Teilnehmer ist disjunkt und die Messung erfolgt in den getrennten Gruppen)."
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    "#Subset erstellen, dabei nicht vorhandene Faktorlevel entfernen\n",
    "sample_data_mw <- droplevels(sample_data[sample_data$Geschlecht != \"d\", ])\n",
    "#Boxplots für die weiblichen und männlichen Probanden in der Stichprobe erzeugen\n",
    "ggplot(sample_data_mw, aes(Geschlecht, Groesse)) + geom_boxplot() \n",
    "#Betrachten der deskriptiven Eckdaten der Größenverteilungen\n",
    "by(sample_data_mw$Groesse, INDICES = sample_data_mw$Geschlecht, FUN = describe)"
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    "Ergänzend kann auch eine Darstellung der Mittelwerte mit den Konfidenzintervallen bereits Aufschluss über Unterschiede in bzgl. der Mittelwerte geben."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "33014828",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Barplot mit Konfidenzintervallen \n",
    "ggplot(sample_data_mw, aes(Geschlecht, Groesse)) +\n",
    "    #ggf. ergänzend Verteilung in einem Violin-Plot unterlegen, um ggf. Besonderheiten der Verteilung sichtbar zu machen\n",
    "    #(Mischung aus Box-Plot und Darstellung der Dichtefunktion)  \n",
    "    #geom_violin() +\n",
    "    #Darstellen der Mittelwerte als Punkte\n",
    "    stat_summary(fun=mean, geom=\"point\") + \n",
    "    #Darstellen der Konfidenzintervalle (Standardfehler für den Mittelwert mit Argument \n",
    "    #für 95% Konfidenzintervall (mult = 1.96))\n",
    "    stat_summary(fun.data = mean_se, geom = \"errorbar\", fun.args = list(mult = 1.96), width = 0.2) "
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   "cell_type": "markdown",
   "id": "bd9d7cf2",
   "metadata": {},
   "source": [
    "Die Funktion zur Durchführung des t-Tests in R heißt `t-test()` und ist vom Aufbau der Übergabeparameter der Funktion `lm()`, die wir für die lineare Regression verwendet haben, recht ähnlich. Zusätzlich verfügt die Methode über einen zusätzlichen Übergabeparameter `paired`, über den festgelegt werden kann, ob ein t-Test für unabhängige Stichproben (`paired=FALSE`) oder für abhängige Stichproben (`paired=TRUE`) verwendet werden soll. Wird keine Angabe gemacht, wird ein t-Test für unabhängige Stichproben durchgeführt.\n",
    "\n",
    "```R\n",
    "    t.test(kriteriumsvariable ~ praediktor, data=data_frame_mit_daten, paired=FALSE)\n",
    "```"
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "497a9fb0",
   "metadata": {},
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   "source": [
    "#Durchführen des t-Test für die ungerichtete Hypothese, dass der Mittelwert der nach Geschlecht getrennten Gruppen\n",
    "#hinsichtlich der Größe einen Unterschied aufweist.\n",
    "ergebnis_t_test <- t.test(Groesse ~ Geschlecht, sample_data_mw);ergebnis_t_test"
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   "id": "db301f91",
   "metadata": {},
   "source": [
    "Das Ergebnis für den t-Test mit der oben aufgestellten, ungerichteten Unterschiedshypothese liefert R folgendes Ergebnis:\n",
    "\n",
    "![t-Test_Ergebnis-2.svg](attachment:t-Test_Ergebnis-2.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a0827aa",
   "metadata": {},
   "source": [
    "In der mit Ziffer 1 gekennzeichneten Zeile sehen Sie die Angaben zur Teststatistik t, die uns auch bereits bei anderen statistischen Tests begegnet sind. Darüber hinaus sind in dieser Zeile die Freiheitsgrade, die durch die Korrektur im Rahmen des Welch Tests Nachkommastellen aufweisen, und der Signifikanzwert ausgewiesen. In der Zeile mit Ziffer 2 finden wir die Alternativhypothese aufgeführt und unter Ziffer 3 das Konfidenzintervall für die Differenzen zwischen den Mittelwerten ausgewiesen. Neben Ziffer 4 finden Sie schließlich die Mittelwerte der beiden Stichproben. Der Test ist in unserem Beispiel hochsignifikant. Das Ergebnis lässt sich wie folgt ausdrücken (siehe z.B. Field 2012, S. 385): Im Mittel sind männliche Probanden ($\\bar{x}$ = 184,7, SE=0.46) größer als weibliche Probanden ($\\bar{x}$ = 166,5, SE = 0.31), t(840,48)=33,08, p < 0,001."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8393e955",
   "metadata": {},
   "source": [
    "Wir haben oben bereits angesprochen, dass ein signifikantes Ergebnis allein noch nichts über die Bedeutung des gefundenen Effekts aussagt. Entsprechend ist auch für den t-Test eine Effektgröße anzugeben. Aus der Teststatistik $t$ lässt sich auf recht einfache Weise der bereits bekannte Effektgröße `r` berechnen. Die Formel für die Berechnung lautet: \n",
    "$$ r = \\sqrt{\\frac{t^2}{t^2 + df}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e44ccaa5",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Umsetzung der Formel in R mit den Werten aus dem durchgeführten t-Test\n",
    "#1. Betrachten der Struktur des Ergebnisobjekts\n",
    "str(ergebnis_t_test)\n",
    "#2. t und df aus dem Ergebnisobjekt des t-Tests extrahieren\n",
    "t <- ergebnis_t_test$statistic[[1]]\n",
    "df <- ergebnis_t_test$parameter[[1]]\n",
    "#3. Einsetzen in obige Formel\n",
    "r <- sqrt(t^2/(t^2+df)); r"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c579093",
   "metadata": {},
   "source": [
    "Die Interpretation der Effektgröße erfolgt analog zur Interpretation bei in den anderen bereits kennengelernten Kontexten. Dabei wird ein Wert von r $\\geq$ 0,5 grundsätzlich als großer Effekt betrachtet, allerdings darf eine Interpretation nie unabhängig vom jeweiligen Forschungskontext stattfinden.\n",
    "\n",
    "Ein weiteres übliches Maß für die Effektgröße -- gerade bei der Betrachtung von Gruppendifferenzen -- stellt das d-Maß von Cohen dar. Die Grenzen für kleine, mittlere und große Effekte liegen bei diesem Maß bei 0,2, 0,5 und 0,8 (vgl. z.B. Döring und Bortz 2016, S. 820). Der Wertebereich ist im Gegensatz zu dem des Korrelationskoeffizienten nicht begrenzt und reicht von $-\\infty$ bis $+\\infty$.\n",
    "\n",
    "Die beiden Effektgrößen lassen sich dabei jeweils wechselseitig umrechnen, wenn die Gruppengrößen näherungsweise gleich groß sind. Für die Umwandlung des Korrelationskoeffizienten $r$ in Cohens $d$ kann die folgende Formel verwendet werden (vgl. z.B. Döring und Bortz 2016, S. 819): \n",
    "\n",
    "$$d = \\frac{2 \\cdot r}{\\sqrt{1-r^2}}$$\n",
    "\n",
    "Für unser Beispiel ergibt sich entsprechend ein gerundeter Wert von 2,28. Auch nach der gängigen Auffassung für diese Effektgröße handelt es sich demnach um einen großen Effekt. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "545ba43e",
   "metadata": {},
   "outputs": [],
   "source": [
    "d = (2 * r)/sqrt(1-r^2);d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a0450f6",
   "metadata": {},
   "source": [
    "Auch Unterschiedshypothesen können als gerichtete Hypothesen aufgestellt werden, wie Sie bereits aus der Lektüre entnehmen konnten. Diese könnte für das obige Beispiel wie folgt lauten: \"Die Größe von Männern und Frauen unterscheidet sich dahingehend, dass Frauen im Mittel kleiner sind als Männer\" (formal: $H_1: \\bar{x}_{Männer} > \\bar{x}_{Frauen}$ oder alternativ: $\\bar{x}_{Männer} - \\bar{x}_{Frauen} > 0$). Die Nullhypothese lautet entsprechend: \"Die Größe von Männern und Frauen unterscheidet sich nicht oder in der Art, dass Frauen im Mittel größer sind als Männer.\" ($H_0: \\bar{x}_{Männer} \\leq \\bar{x}_{Frauen}$ oder alternativ $\\bar{x}_{Männer} - \\bar{x}_{Frauen} \\leq 0$). Letztere Schreibweise vereinfacht die korrekte Angabe der Richtung des einseitigen Signifikanztests in R. Dabei ist darauf zu achten, dass die Reihenfolge der Mittelwerte mit der Kodierung der Faktorvariable identisch ist.\n",
    "\n",
    "Für die Prüfung gerichteter Hypothesen stehen bei der Funktion `t.test()` ähnlich den Optionen der Funktion `cor.test()` für den Übergabeparameter `alternative` die Optionen `less` und `greater` zur Wahl. Wird keine Einstellung über diesen Parameter vorgenommen, wird davon ausgegangen, dass ein zweiseitiger Signifikanztest (`two.sided`) durchgeführt werden soll. Die Option `less` verwendet die Nullhypothese, dass die Differenz zwischen den Mittelwerten kleiner als 0 ist, die Option `greater` entsprechend, dass die Differenz größer als 0 ist. Dabei spielt die Reihenfolge der Faktorlevel des Prädiktors wie oben erwähnt eine entscheidende Rolle. In unserem Beispiel ist die Variable Geschlecht so kodiert, dass `m` die Ziffer 1 und `f` die Ziffer 2 zugewiesen ist, die Reihenfolge bei den Faktorleveln also `m` vor `f` entspricht. Die Alternativhypothese ist daher für den t-Test in R formal gemäß dieser Reihenfolge zu formulieren: $\\bar{x}_{Männer} - \\bar{x}_{Frauen} > 0$\n",
    "\n",
    "In den beiden nachfolgenden Codezellen wird der Einfluss der Kodierung demonstriert, indem in der zweiten Zelle die Reihenfolge der Faktorlevel für das Geschlecht umgekehrt wird. Dort wird dann die Option `less` für den einseitigen Hypothesentest angegeben. Beide Berechnungen zeigen das gleiche Ergebnis. Wenn Sie bei der zweiten Variante die Option `greater` wählen, liegt der Signifikanzwert bei 1 und die Nullhypothese kann nicht abgelehnt werden. Welche Reihenfolge der Gruppen im Test angenommen wird, kann auch indirekt durch die Reihenfolge in der Ausgabe der Mittelwerte ermittelt werden. Diese sind in den Ergebnisausgaben der beiden t-Tests jeweils umgekehrt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3ac3edc5",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Reihenfolge der Faktorvariable für das Geschlecht prüfen, \n",
    "#um den Übergabewert für den Parameter 'alternative' korrekt zu spezifizieren.\n",
    "levels(sample_data_mw$Geschlecht) #Reihenfolge der Level: m, w (--> x_m - x_w > 0: greater)\n",
    "t.test(Groesse ~ Geschlecht, sample_data_mw, alternative = \"greater\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c4445741",
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_data_mw$Geschlecht <- factor(sample_data_mw$Geschlecht, levels = c(\"w\", \"m\"))\n",
    "levels(sample_data_mw$Geschlecht) #Reihenfolge der Level: w, m (--> x_w - x_m < 0: less)\n",
    "t.test(Groesse ~ Geschlecht, sample_data_mw, alternative = \"less\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c59e31eb",
   "metadata": {},
   "outputs": [],
   "source": [
    "t.test(Groesse ~ Geschlecht, sample_data_mw)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7a6fc9e",
   "metadata": {},
   "source": [
    "##### Aufgabe<a class=\"tocSkip\">\n",
    "Stellen Sie eine Unterschiedshypothese im Zusammenhang mit den Geschlechtern \"männlich\" und \"weiblich\" für die Größe auf und überprüfen Sie diese mit einem t-Test. Berechnen Sie auch die Effektgröße und wählen Sie eine geeignete graphische Darstellungsform zur Veranschaulichung der Ergebnisse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7f3ed660",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "fe075e67",
   "metadata": {},
   "source": [
    "Sie haben nun einen ersten Einblick in das Testen von Unterschiedshypothesen erhalten. Neben dem t-Test sind für solche Untersuchungen insbesondere die Varianzanalysen (ANOVA) zu nennen, die wir im Kontext dieses Kurses aber nicht mehr behandeln können. Bei Interesse finden Sie eine schöne Einführung in den beiden begleitenden Werken, die als Grundlagenlektüre für diesen Kurs ausgewiesen sind."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc777140",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "# Quellen und Literatur\n",
    "- Bortz, Jürgen; Schuster, Christof (2010): Statistik für Human- und Sozialwissenschaftler. Mit … 163 Tabellen. 7., vollst. überarb. und erw. Aufl. Berlin: Springer (Springer-Lehrbuch).\n",
    "- Döring, Nicola; Bortz, Jürgen (2016): Forschungsmethoden und Evaluation in den Sozial- und Humanwissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.\n",
    "- Field, Andy; Miles, Jeremy; Field, Zoë (2012): Discovering statistics using R. London: SAGE."
   ]
  },
  {
   "attachments": {
    "Stichprobenverteilung.png": {
     "image/png": 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    "# Fußnoten\n",
    "<a href=\"#footnote1\"><sup>1</sup></a><span id=\"footnote1\"> Den Begriff Stichprobenverteilung (engl. sampling distribution) sollten Sie bereits aus Field et al. (2012), Kapitel 2 kennen: Werden aus einer Population verschiedene Stichproben gezogen und jeweils dieselbe statistische Kennzahl (z.B. das arithmetische Mittel) berechnet, so weichen diese i.d.R. voneinander ab. Bei der Stichprobenverteilung handelt es sich nun um die Häufigkeitsverteilung der ermittelten statistischen Kennzahlen. Die Standardabweichung der Stichprobenmittelwerte wird als Standardfehler bezeichnet und stellt ein Maß für die Repräsentativität der Stichprobe in Bezug auf die Grundgesamtheit dar.</span> [&#9650;](#Normalverteilung-der-Daten)\n",
    "![Stichprobenverteilung.png](attachment:Stichprobenverteilung.png)\n",
    "\n",
    "<a href=\"#footnote2\"><sup>2</sup></a><span id=\"footnote2\"> In der Literatur wird vielfach ein Stichprobenumfang von mindestens 30 Probanden benannt, um von einer großen Stichprobe zu sprechen.</span> [&#9650;](#Normalverteilung-der-Daten)\n",
    "\n",
    "<a href=\"#footnote3\"><sup>3</sup></a><span id=\"footnote3\"> Die Anzahl frei variierbarer Werte bei der Berechnung einer statistischen Kennzahl oder einem statistischen Test. Ausführliche Erläuterungen finden Sie bei Field et al. (2012) in Kapitel 2 oder bei Döring und Bortz (2016) in Abschnitt 12.4.</span> [&#9650;](#Freiheitsgrade)\n",
    "\n",
    "<a href=\"#footnote4\"><sup>4</sup></a><span id=\"footnote4\">Das Konfidenzintervall kennen Sie bereits aus Field et al. (2012), Kapitel 2 und Döring und Bortz (2016) Abschnitt 12.5. Es umspannt einen Wertebereich innerhalb dessen sich der wahre Wert der Grundgesamtheit höchstwahrscheinlich befindet. Typischer Weise werden Konfidenzintervalle für die Wahrscheinlichkeiten von 95% oder 99% berechnet. Je höher die Wahrscheinlichkeit ist, mit der der Populationsparameter innerhalb des Konfidenzintervalls liegt, desto größer ist das Konfidenzintervall naturgemäß.</span> [&#9650;](#Korrelationsanalyse)\n",
    "\n",
    "<a href=\"#footnote5\"><sup>5</sup></a><span id=\"footnote5\">Die Tilde findet sich auf der Tastatur auf derselben Taste wie das `+`-Zeichen. Sie müssen zusätzlich die rechte Alt-Taste (Windows: `AltGr`) gedrückt halten, um das Zeichen zu erhalten.</span> [&#9650;](#regressionsmodell)\n",
    "\n",
    "<a href=\"#footnote5\"><sup>6</sup></a><span id=\"footnote6\">Da die Messwerte bei in den verschiedenen Gruppen von unterschiedlichen Personen stammen, sollte diese Voraussetzung i.d.R. erfüllt sein. Ein Szenario, in dem dies nicht erfüllt wäre, ließe sich konstruieren, indem die Personen der einen Gruppe bei der Untersuchung der anderen Gruppe zuschauen dürften und die abhängige Variabel dadurch beeinflusst würde. Auch bei Ehe- oder Geschwisterpaaren in unterschiedlichen Gruppen kann von einer vollständigen Unabhängigkeit nicht ausgegangen werden.</span> [&#9650;](#t-test-voraussetzungen)"
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