From 941160fa2c1ac8d0132ff25b0b411fe8ca755a20 Mon Sep 17 00:00:00 2001 From: Lukas Bruelheide Date: Thu, 1 Feb 2024 17:40:59 +0100 Subject: [PATCH] regression: continue review, still not finished. Left at lokaltest --- inst/tutorials/5d_chi/vl8_a_chiquadrat.Rmd | 7 +- .../6b_regression/images/duh-prince.gif | Bin 0 -> 2062744 bytes .../6b_regression/vl9_a_regression.Rmd | 91 ++++++++++++++---- 3 files changed, 77 insertions(+), 21 deletions(-) create mode 100644 inst/tutorials/6b_regression/images/duh-prince.gif diff --git a/inst/tutorials/5d_chi/vl8_a_chiquadrat.Rmd b/inst/tutorials/5d_chi/vl8_a_chiquadrat.Rmd index b584743..45cb43a 100644 --- a/inst/tutorials/5d_chi/vl8_a_chiquadrat.Rmd +++ b/inst/tutorials/5d_chi/vl8_a_chiquadrat.Rmd @@ -15,8 +15,11 @@ resource_files: ```{r setup, include=FALSE} library(learnr) -library(tidyverse) knitr::opts_chunk$set(echo = FALSE) ``` -test \ No newline at end of file +```{r question} +question_radio("$p < \\alpha$", + answer("1", correct = T, message = "$p < \alpha$")) +``` + diff --git a/inst/tutorials/6b_regression/images/duh-prince.gif b/inst/tutorials/6b_regression/images/duh-prince.gif new file mode 100644 index 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zp^6y(Y_aHXle}pvVJKbN9UKz&h9k^g4-}JeU+<>Wh@)NF^L)g+q;Xe-%2D-S!oW2(|8A3k(8lS%vexz#>) zTH@)qO&Zp+F4!zK7d~wp!eI`M2ETfB5`m zKt%4`vn%x&s3__}TYs~)Pj}+q-ycLUL(a$o4XUKs#RDmH3a&PYU^3f8=z6qkOKJNhS6? z@QE|JHdKBtT(hy3)pp~()CBX&j#+}%Weip*r1G4PTAgj@6V+dK82s0tQk~I%TM9?B z(s7gzB&(@kr)S;Vk~%f{B&%!l^a?wdnbRAn`+zl2gmi3=?R45OcOnfO$#o>*YO=4j z$`|2S{H0S)?ce!`2)y>6_kqXW`6)aml9n(P#F5QAnMkRzb4Y~#q`9{tdbh{no2A)5 znys9QwV|zrW>vbsJM>X#QZqbFKR>(jIrc|W>QI9ZG}qbtCp4yy+owUfa?frz8=bK; zB@-zTIfL-QL-`DlaqY$&=Fp$dD-JtS)Fiha1?ljimll(Lj`2dH1eE(({Qa_P=bQ~D z`TJZSq?mU*uKQ6WIR{JT`0&468DiIvJi@`GnfW<3(#GCPc}0ed(uvYLPU#$cCdeGQ IU;se #### Was sind $b_0$ und $b_1$? @@ -527,6 +528,7 @@ wenn sie sehr groß sind. **Kleine Residuen bedeuten, dass das Modell näher an #### Formel Ein Residuum $e$ (wie Error) der Messung $i$ errechnet sich aus: + $$e_i = y_i - \hat y_i$$ dabei gilt: @@ -796,7 +798,7 @@ Stichprobe als Faustregel ungefähr $n > 30$. Manche Daten werden besser nicht durch Geraden erklärt, sondern haben vielleicht quadratische Zusammenhänge, wie in diesem Beispiel: -```{r} +```{r sim_nonlinear_data} data <- data.frame(temperatur=c(6, 9, 12, 14, 30, 35, 40, 47, 51, 55, 60), enzymaktivität=c(14, 28, 50, 70, 89, 94, 90, 75, 59, 44, 27)) @@ -986,7 +988,7 @@ Vergleichen wir dafür noch einmal zwei Plots: -```{r, echo=T, message=FALSE, warning=FALSE, setup = "homoskedasis_grafiken"} +```{r grafiken_homosked, echo=T, message=FALSE, warning=FALSE, setup = "homoskedasis_grafiken"} fit1 <- lm(y ~ x, data = df1) plot(fit1, which = 1) @@ -1049,15 +1051,28 @@ h4 + geom_smooth(method = "lm", se = F) + ``` -```{r, echo = TRUE} +```{r durbinwatsonfit, echo = TRUE} car::durbinWatsonTest(fit4) ``` -```{r autocor_question} -question_radio("Was ist deine Hypothesenentscheidung, wenn du den $p$-Wert des Tests mit $\alpha$ = 0.05 vergleichst?", +```{r autocor_question1} +question_radio("Was ist deine Hypothesenentscheidung, wenn du den $p$-Wert des Tests mit $\\alpha = 0.05$ vergleichst?", answer("H0 beibehalten", message = "Da $p$ < $alpha$ ist der Test signifikant. Bei Signifikanz muss die H0 abgelehnt werden."), answer("H0 ablehnen", correct = T, - message = "Da $p$ < $\alpha$, ist der Test signifikant."), + message = "Da $p < \\alpha$, ist der Test signifikant."), + allow_retry = T, + random_answer_order = T, + correct = random_praise("de"), + incorrect = random_encouragement("de")) +``` + + + +```{r autocor_question2} +question_radio("Was bedeutet eine Entscheidung gegen die $H_0$ inhaltlich?", + answer("Es besteht Autokorrelation", correct = T, message = "Da $p$ < $alpha$ ist der Test signifikant. Bei Signifikanz muss die $H_0$ abgelehnt werden. Sie sagt, es gibt keine Autokorrelation. Wenn wir das ablehnen, dann gibt es demzufolge welche."), + answer("Es besteht keine Autokorrelation", + message = "Die $H_0$ besagt genau das, aber wenn wir sie ablehnen gilt das eben nicht."), allow_retry = T, random_answer_order = T, correct = random_praise("de"), @@ -1394,7 +1409,7 @@ Fall also als Volumen (m³). ### $b_0$ (Intercept) Generell gilt: -*„$b_0$ ist der vorhergesagte Wert, wenn der Prädiktor den Wert 0 +*„$b_0$ ist der vorhergesagte Wert des Kriteriums, wenn der Prädiktor den Wert 0 annimt."* Bezogen auf unser Beispiel: @@ -1410,13 +1425,36 @@ so dass $x = 0$ einen sinnvollen Wert darstellt. Siehe den Exkurs: "Zentrieren". ### $b_1$ (Steigung) Generell: -*„$b_1$ ist die vorhergesagte Änderung, wenn der Prädiktor um **eine Einheit** +*„$b_1$ ist die vorhergesagte Änderung im Kriterium, wenn der Prädiktor um **eine Einheit** erhöht wird."* Für unser Beispiel: *„Pro Zentimeter Durchmesser mehr steigt unser erwartetes Volumen um 0.0565 Kubikmeter."* +Erinnert euch an Schulmathe, die Gerade und das Steigungsdreieck: Wenn man $x$ um eine Einheit erhöht, wie viel größer wird dann $y$? Das ist die Steigung der Geraden. + +```{r steigungsdreieck} +pred <- function(x) -1.05 + x*0.05605 +yend <- pred(5) +ystart <- pred(4) +ggplot() + + stat_function(fun = `.gerade`, xlim = c(-1, 22), args = list(slope = 0.05605, intercept = -1.05)) + + geom_segment(aes(x = 4, y = pred(4), yend = pred(4), xend = 5), color = "red", + arrow = arrow(length = unit(0.03, "npc"))) + + geom_segment(aes(x = 5, xend = 5, y = !!pred(4), yend = !!pred(5)), color = "red", + arrow = arrow(length = unit(0.03, "npc"))) + + annotate("text", x = 4.5, y = pred(3.4), label = "1 cm") + + annotate("text", x = 5.8, y = pred(4.5), label = "0.0565 m³") + + coord_cartesian(xlim = c(0, 10), ylim = c(-1.06, -0.4), expand = F) + + scale_y_continuous(breaks = scales::breaks_pretty()) + + scale_x_continuous(breaks = 0:10) + + labs(y = "y: Vorherges. Holzvol. (m³)", x = "x: Durchmesser (cm)") + + theme_minimal() +``` + +Schön zu sehen ist auf der Grafik auch der Achsenabschnitt (*Intercept*) $b_0$ bei $x = 0$. + ::: gelb Das Regressionsgewicht $b_1$ ist eng verwandt mit der Korrelation, aber es gibt einen wichtigen Unterschied: Es ist nicht „symmetrisch“ wie eine @@ -1504,6 +1542,7 @@ höher, außer man korrigiert diese Verzerrung. Das heißt dann einfache lineare Regression machen, also nur einen Prädiktor verwenden. ::: +
    #### Interpretation @@ -1540,7 +1579,7 @@ Im Fall der einfachen linearen Regression, aber nicht bei multipler Regression, entspricht $R^2$ der quadrierten Pearson-Korrelation $r$ des Prädiktors mit dem Kriterium. Daraus folgt: $\sqrt{R^2} = r$ -#### Übung +**Übung** ::: aufgabe Berechnen Sie den Korrelationskoeffizienten auf zwei Wegen: @@ -1580,7 +1619,7 @@ question_numeric("Geben Sie Ihren errechneten Wert der Pearson-Korrelation zwisc allow_retry = TRUE) ``` -#### Reminder: Interpretation Korrelation +**Reminder: Interpretation Korrelation** Wie interpretiert man noch mal den Korrelationskoeffizienten? Siehe Tutorial "Korrelation". @@ -1640,10 +1679,11 @@ eigentlich generell im ganzen Wald nicht der Fall ist. ![](images/ftest.jpg){width="60%"} -Die Nullhypothese $H_0$ des Gesamtmodells lautet daher: +Die Hypothesen des Gesamtmodelltests lauten daher: + +$H_0: R^2 = 0$ -$H_0 = R^2$ beträgt in Wahrheit 0. -$H_1 = R^2$ ist ungleich 0. +$H_1: R^2 \neq 0$ ### R-Output @@ -1684,13 +1724,25 @@ learnr::question_radio("Wie würde darauf basierend deine Hypothesenentscheidung ) ``` -### APA Bericht +```{r gesamtmodelltest_inhaltsfrage} +question_radio("Was bedeutet eine Entscheidung gegen die $H_0$ inhaltlich?", + answer("Das Modell leistet einen Beitrag zur Aufklärung der Varianz", correct = T, message = "Da $p$ < $alpha$ ist der Test signifikant. Bei Signifikanz muss die $H_0$ abgelehnt werden."), + answer("Das Modell leistet keinen Beitrag zur Aufklärung der Varianz", + message = "Die $H_0$ besagt genau das, aber wenn wir sie ablehnen gilt das eben nicht."), + allow_retry = T, + random_answer_order = T, + correct = random_praise("de"), + incorrect = random_encouragement("de")) +``` + + +**APA-Bericht** Super! Wir können also davon ausgehen, dass: ::: blau-nb *"Der Durchmesser von Kirschbäumen erklärt einen signifikanten Anteil der Varianz -des Volumens dieser Bäume $(F(1, 29) = 419.4, p < .001)$."* +des Volumens dieser Bäume $(R^2 = .935,\ F(1, 29) = 419.4,\ p < .001)$."* ::: ### Lokaltest @@ -1699,10 +1751,11 @@ Der Lokaltest testet jeden Regressionskoeffizienten einzeln (und ist deswegen besonders für die multiple lineare Regression wichtig, wo es mehrere Prädiktoren gibt). -Im Fall der einfachen linearen Regression entspricht der Signifikanztest -des Lokaltest dem Test für das Gesamtmodell, weil das gesamte Modell nur aus einem +Im Fall der einfachen linearen Regression entspricht der Lokaltest dem Test für das Gesamtmodell, weil das gesamte Modell nur aus einem Prädiktor besteht. +![](images/duh-prince.gif){width="40%"} + Das lässt sich auch schön zeigen anhand der Beziehung zwischen der $t$-Statistik aus dem Lokaltest und der $F$-Statistik aus dem Gesamtmodelltest: @@ -1829,7 +1882,7 @@ gut die Gerade die Datenpunkte "erklärt". ### Der schnelle Weg mit base R -```{r, echo = TRUE} +```{r fitvis, echo = TRUE} fit <- lm(volume ~ diameter, data = trees) # Modell erstellen plot(volume ~ diameter, data = trees) # Punktdiagramm erstellen @@ -1916,7 +1969,7 @@ abline(temp_model) # Regressionsgerade zum Plot hinzufügen Und hier der zweite Weg mit `ggplot`: -```{r, echo = TRUE} +```{r ggplot_beauty, echo = TRUE} library(ggplot2) # Paket laden ggplot(trees, aes(x = diameter, y = volume)) + # Daten und Mappings definieren

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