class: center, middle, inverse, title-slide .title[ # <b>Week 11: Some Kind of End to the Course</b> ] .subtitle[ ## Univariate Statistics and Methodology using R ] .author[ ### ] .institute[ ### Department of Psychology<br/>The University of Edinburgh ] --- class: inverse, center, middle # Part 1 ## Week 7 - 10 Recap --- # Describing a pattern with a line .pull-left[  ] .pull-right[ ```r ggplot(df, aes(x = x1, y = y)) + geom_point() ``` ] --- # Defining a line .pull-left[ <!-- --> ] .pull-right[ `$$\Large y_i = b_0 + b_1(x_{1i}) + \varepsilon_i$$` A line can be defined by two values: - A starting point (Intercept) - A slope ($y$ across `\(x1\)`) Fitting a linear model to some data provides coefficient estimates `\(\hat{b}_0\)` and `\(\hat{b}_1\)` that minimise `\(\sigma_\varepsilon\)`. ] --- # Testing the coefficients (1) .pull-left[ <!-- --> ] .pull-right[ `$$\Large \hat{y} = \color{red}{\hat{b}_0} + \hat{b}_1(x_{1})$$` In the "null universe" where `\(b_0 = 0\)`, when sampling this many people, what is the probability that we will find an intercept at least as extreme as the one we _have_ found? ``` ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 6.982 1.026 6.81 1.5e-08 *** ## x1 0.806 0.234 3.45 0.0012 ** ``` ] --- # Testing the coefficients (2) .pull-left[ <!-- --> ] .pull-right[ `$$\Large \hat{y} = \hat{b}_0 + \color{red}{\hat{b}_1}(x_{1})$$` In the "null universe" where `\(b_1 = 0\)`, when sampling this many people, what is the probability that we will find a relationship at least as extreme as the one we _have_ found? ``` ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 6.982 1.026 6.81 1.5e-08 *** *## x1 0.806 0.234 3.45 0.0012 ** ``` ] --- # Coefficient Sampling Variability .pull-left[ <!-- --> ] .pull-right[ `$$\Large \hat{y} = \hat{b}_0 + \hat{b}_1(x_{1})$$` Plausible range of values for `\(b_0\)` and `\(b_1\)`: ```r confint(model) ``` ``` ## Estimate 2.5 % 97.5 % ## (Intercept) 6.9816 4.919 9.044 ## x1 0.8057 0.336 1.275 ``` ] --- # Testing the model <!-- --> .pull-left[ $$ \small `\begin{align} & F_{df_{model},df_{residual}} = \frac{MS_{Model}}{MS_{Residual}} = \frac{SS_{Model}/df_{Model}}{SS_{Residual}/df_{Residual}} \\ & df_{model} = \text{nr predictors} \\ & df_{residual} = \text{sample size} - \text{nr predictors} - 1 \\ \end{align}` $$ ] .pull-right[ ``` ## ## Multiple R-squared: 0.199, Adjusted R-squared: 0.182 *## F-statistic: 11.9 on 1 and 48 DF, p-value: 0.00118 ``` ] --- # More predictors .pull-left[ <!-- --> ] .pull-right[ <!-- --> ] --- # More predictors (2) <br><br> `$$\Large y_i = b_0 + b_1(x_{1i}) + b_2(x_{2i}) + \varepsilon_i$$` - A starting point (Intercept) - A slope (across `\(x_1\)`) - _Another_ slope (across `\(x_2\)`) Coefficient estimates `\(\hat{b}_0\)`, `\(\hat{b}_1\)`, `\(\hat{b}_2\)` minimise `\(\sigma_\varepsilon\)`. --- # More predictors (3) <!-- --> --- # _Even_ more predictors... <br><br> `$$\Large y_i = b_0 + b_1(x_{1i}) + b_2(x_{2i}) + \, ... \, + b_k(x_{ki}) + \varepsilon_i$$` - A starting point (Intercept) - A slope (across `\(x_1\)`) - A slope (across `\(x_2\)`) - ... - ... - A slope (across `\(x_k\)`) --- # associations that depend on other things .pull-left[ <!-- --> ] .pull-right[ <!-- --> ] --- # interactions <br><br> `$$\Large y_i = b_0 + b_1(x_{1i}) + b_2(x_{2i}) + b_3(x_{1i}\cdot x_{2i}) + \varepsilon_i$$` - starting point (Intercept) - A slope (across `\(x_1\)`) - A slope (across `\(x_2\)`) - ... - How slope across `\(x_1\)` changes across `\(x_2\)` --- # interactions .pull-left[ <!-- --> ] .pull-right[ ] --- # interactions (2) .pull-left[ <!-- --> ] .pull-right[ <!-- --> ] --- # other outcomes .pull-left[ `$$ln(\frac{p}{1-p}) = b_0 + b_1(x_{1i}) + b_2(x_{2i})$$` <!-- --> ] .pull-right[ $$ ln(\frac{p}{1-p}) \, \Rightarrow \, \frac{p}{1-p} \, \Rightarrow \, p $$ <!-- --> ] --- # Checking Assumptions: Linear Models .pull-left[ ### required - linearity of relationship - for the _residuals_: + normality + homogeneity of variance + independence ] .pull-right[ ### desirable - uncorrelated predictors - no "bad" (overly influential) observations ] --- # Checking Assumptions: Logit Models .pull-left[ ### required - linearity of relationship .red[between IVs and log-odds] - for the _residuals_: + .red[~~normality~~] + .red[~~homogeneity of variance~~] + independence ] .pull-right[ ### desirable - uncorrelated predictors - no "bad" (overly influential) observations - .red[large samples (due to maximum likelihood fitting)] ] --- class: inverse, center, middle, animated, flipInY # End of Part 1 --- class: inverse, center, middle # Part 2 ## Common Tests as linear models ```r usmr <- read_csv("https://uoepsy.github.io/data/usmr2022.csv") ``` --- # lm vs correlation .pull-left[ #### regression, continuous predictor ```r summary(lm(sleeprating ~ height, data = usmr)) ``` ``` ## ## Call: ## lm(formula = sleeprating ~ height, data = usmr) ## ## Residuals: ## Min 1Q Median 3Q Max ## -65.42 -11.66 5.52 16.96 36.16 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 18.785 46.879 0.4 0.69 ## height 0.279 0.278 1.0 0.32 ## ## Residual standard error: 22.6 on 76 degrees of freedom ## (2 observations deleted due to missingness) ## Multiple R-squared: 0.013, Adjusted R-squared: 4.73e-05 ## F-statistic: 1 on 1 and 76 DF, p-value: 0.32 ``` ] .pull-right[ #### Correlation ```r cor.test(usmr$height, usmr$sleeprating) ``` ``` ## ## Pearson's product-moment correlation ## ## data: usmr$height and usmr$sleeprating ## t = 1, df = 76, p-value = 0.3 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## -0.1112 0.3284 ## sample estimates: ## cor ## 0.1142 ``` ] --- # lm vs t.test .pull-left[ #### regression, intercept ```r summary(lm(height ~ 1, data = usmr)) ``` ``` ## ## Call: ## lm(formula = height ~ 1, data = usmr) ## ## Residuals: ## Min 1Q Median 3Q Max ## -15.407 -7.607 -0.107 7.523 20.893 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 168.11 1.04 162 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 9.22 on 78 degrees of freedom ## (1 observation deleted due to missingness) ``` ] .pull-right[ #### one sample t.test ```r t.test(usmr$height, mu=0) ``` ``` ## ## One Sample t-test ## ## data: usmr$height ## t = 162, df = 78, p-value <2e-16 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 166.0 170.2 ## sample estimates: ## mean of x ## 168.1 ``` ] --- # lm vs t.test (2) .pull-left[ #### regression, binary predictor ```r summary(lm(height ~ catdog, data = usmr)) ``` ``` ## ## Call: ## lm(formula = height ~ catdog, data = usmr) ## ## Residuals: ## Min 1Q Median 3Q Max ## -13.655 -7.501 0.499 8.399 19.499 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 166.36 1.55 107.60 <2e-16 *** ## catdogdog 3.15 2.07 1.52 0.13 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 9.15 on 77 degrees of freedom ## (1 observation deleted due to missingness) ## Multiple R-squared: 0.0291, Adjusted R-squared: 0.0165 ## F-statistic: 2.31 on 1 and 77 DF, p-value: 0.133 ``` ] .pull-right[ #### two sample t.test ```r t.test(height ~ catdog, data = usmr, var.equal = TRUE) ``` ``` ## ## Two Sample t-test ## ## data: height by catdog ## t = -1.5, df = 77, p-value = 0.1 ## alternative hypothesis: true difference in means between group cat and group dog is not equal to 0 ## 95 percent confidence interval: ## -7.2706 0.9796 ## sample estimates: ## mean in group cat mean in group dog ## 166.4 169.5 ``` ] --- # lm vs Traditional ANOVA > If you should say to a mathematical statistician that you have discovered that linear multiple regression and the analysis of variance (and covariance) are identical systems, he would mutter something like "Of course&mdash;general linear model," and you might have trouble maintaining his attention. If you should say this to a typical psychologist, you would be met with incredulity, or worse. Yet it is true, and in its truth lie possibilities for more relevant and therefore more powerful research data. .tr[ Cohen (1968) ] --- # History .pull-left[ .br3.pa2.bg-gray.white[ ### .white[Multiple Regression] - introduced c. 1900 in biological and behavioural sciences - aligned to "natural variation" in observations - tells us that means `\((\bar{y})\)` are related to groups `\((g_1,g_2,\ldots,g_n)\)` ]] .pull-right[ .br3.pa2.bg-gray.white[ ### .white[ANOVA] - introduced c. 1920 in agricultural research - aligned to experimentation and manipulation - tells us that groups `\((g_1,g_2,\ldots,g_n)\)` have different means `\((\bar{y})\)` ]] .pt2[ - both produce `\(F\)`-ratios, discussed in different language, but identical ] --- # lm vs Traditional ANOVA .pull-left[ #### regression, binary predictor ```r summary(lm(height ~ eyecolour, data = usmr)) ``` ``` ## ## lm(formula = height ~ eyecolour, data = usmr) ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 170.254 2.183 77.99 <2e-16 *** ## eyecolourbrown -3.682 2.609 -1.41 0.16 ## eyecolourgreen 2.717 4.125 0.66 0.51 ## eyecolourgrey -0.254 9.515 -0.03 0.98 ## eyecolourhazel -2.404 3.935 -0.61 0.54 ## eyecolourother -4.834 5.775 -0.84 0.41 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 9.26 on 73 degrees of freedom ## (1 observation deleted due to missingness) ## Multiple R-squared: 0.0563, Adjusted R-squared: -0.00837 ## F-statistic: 0.871 on 5 and 73 DF, p-value: 0.505 ``` ] .pull-right[ #### anova ```r summary(aov(height ~ eyecolour, data = usmr)) ``` ``` ## Df Sum Sq Mean Sq F value Pr(>F) ## eyecolour 5 373 74.7 0.87 0.51 ## Residuals 73 6261 85.8 ## 1 observation deleted due to missingness ``` ] --- # Why Teach LM/Regression? - LM has less restrictive assumptions + especially true for unbalanced designs/missing data - LM is far better at dealing with covariates + can arbitrarily mix continuous and discrete predictors - LM is the gateway to other powerful tools + mixed models and factor analysis (→ MSMR) + structural equation models --- class: inverse, center, middle, animated, flipInY # End --- background-image: url(lecture_11_files/img/playmo_goodbye.jpg) background-size: contain # Goodbye!