class: center, middle, inverse, title-slide # <b>Week 8: The Linear Model (ctd)</b> ## Univariate Statistics and Methodology using R ### Martin Corley ### Department of Psychology<br/>The University of Edinburgh --- class: inverse, center, middle # Part 1 ## Checking Assumptions --- # Back on the Road Again! .pull-left[ <!-- --> .tc[ "for every extra 0.01% blood alcohol, reaction time slows down by around 32 ms" ] ] .pull-right[  ] --- # The Model We Nearly Made ### All 50 Participants ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = dat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -115.92 -40.42 1.05 42.93 126.64 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 321 91 3.53 0.00093 *** ## BloodAlc 3228 888 3.64 0.00067 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 55.8 on 48 degrees of freedom ## Multiple R-squared: 0.216, Adjusted R-squared: 0.2 ## F-statistic: 13.2 on 1 and 48 DF, p-value: 0.000673 ``` ??? - note the _huge_ affect of `BloodAlc` - why is this? [pause?] - because these are "real" units, not units x 100 --- # Assumptions of Linear Models .pull-left[ .br3.bg-gray.white.pa2[ ### required - **linearity** of relationship(!) - for the _residuals_: + **normality** + **homogeneity of variance** + **independence** ] ] .pull-right[ .br3.bg-gray.white.pa2[ ### desirable <!-- - uncorrelated predictors (no collinearity) --> - no 'bad' (overly influential) observations ]] --- # Residuals .pull-left[ `$$y_i=b_0+b_1\cdot{}x_i+\epsilon_i$$` `$$\color{red}{\epsilon\sim{}N(0,\sigma)~\text{independently}}$$` - normally distributed (mean should be `\(\simeq{}\)` zero) - homogeneous (differences from `\(\hat{y}\)` shouldn't be systematically smaller or larger for different `\(x\)`) - independent (residuals shouldn't influence other residuals) ] .pull-right[ <!-- --> ] --- # At A Glance ```r summary(mod) ``` ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = dat) ## ## Residuals: ## Min 1Q Median 3Q Max *## -115.92 -40.42 1.05 42.93 126.64 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 321 91 3.53 0.00093 *** ## BloodAlc 3228 888 3.64 0.00067 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 55.8 on 48 degrees of freedom ## Multiple R-squared: 0.216, Adjusted R-squared: 0.2 ## F-statistic: 13.2 on 1 and 48 DF, p-value: 0.000673 ``` --- # In More Detail .pull-left[ ### linearity ```r plot(mod,which=1) ``` - plots residuals `\(\epsilon_i\)` against fitted values `\(\hat{y}_i\)` - the 'average residual' is roughly zero across `\(\hat{y}\)`, so relationship is likely to be linear ] .pull-right[  ] --- # In More Detail .pull-left[ ### normality ```r hist(resid(mod)) ``` ] .pull-right[  ] --- count: false # In More Detail .pull-left[ ### normality ```r plot(density(resid(mod))) ``` - check that residuals `\(\epsilon\)` are approximately normally distributed - in fact there's a better way of doing this... ] .pull-right[  ] --- name: QQ # In More Detail .pull-left[ ### normality ```r plot(mod,which=2) ``` - **Q-Q plot** compares the residuals `\(\epsilon\)` against a known distribution (here, normal) - observations close to the straight line mean residuals are approximately normal - numbered observations refer to _row numbers_ in the original data, for checking ] .pull-right[  ] --- # Q-Q Plots .pull-left[ #### y axis - for a normal distribution, what values _should_ (say) 2%, or 4% of the observations lie below? - expressed in "standard deviations from the mean" ```r qnorm(c(.02,.04)) ``` ``` ## [1] -2.054 -1.751 ``` ] -- .pull-right[ #### x axis - from our residuals, what values _are_ (say) 2%, or 4%, of observations found to be less than? - convert to "standard deviations from the mean" ```r quantile(scale(resid(mod)),c(.02,.04)) ``` ``` ## 2% 4% ## -1.527 -1.466 ``` ] -- - Q-Q Plot shows these values plotted against each other --- template: QQ --- # In More Detail .pull-left[ ### homogeneity of variance ```r plot(mod,which=3) ``` - the _size_ of the residuals is approximately the same across values of `\(\hat{y}\)` ] .pull-right[  ] --- # Visual vs Other Methods - statistical ways of checking assumptions will be introduced in labs - they tend to have limitations (for example, they're susceptible to sample size) - nothing beats looking at plots like these (and `plot(<model>)` makes it easy) - however, two things: -- .pt2[ 1. there are no criteria for deciding exactly when assumptions are sufficiently met + it's a matter of experience and judgement 1. we need to talk about **independence** of residuals ] --- class: inverse, center, middle, animated, flip # End of Part 1 --- class: inverse, center, middle # Part 2 ## Independence, Influence --- # Independence - no easy way to check independence of residuals - in part, because it depends on the _source_ of the observations .pt2[ - one determinant might be a single person being observed multiple times - e.g., my reaction times might tend to be slower than yours<br/> `\(\rightarrow\)` multivariate statistics ] --- # Independence - another determinant might be _time_ - observations in a sequence might be autocorrelated - can be checked using the Durbin-Watson Test from the `car` package ```r library(car) dwt(mod) ``` ``` ## lag Autocorrelation D-W Statistic p-value *## 1 -0.1377 2.22 0.42 ## Alternative hypothesis: rho != 0 ``` - shows no autocorrelation at lag 1 --- # Influence .center[ <!-- --> ] - even substantial **outliers** may only have small effects on the model - here, only the intercept is affected --- # Influence .center[ <!-- --> ] - observations with high **leverage** are inconsistent with other data, but may not be distorting the model --- # Influence .center[ <!-- --> ] - we care about observations with high **influence** (outliers with high leverage) --- # Cook's Distance .center[ <!-- --> ] - a standardised measure of "how much the model differs without observation `\(i\)`" --- .flex.items-center[.w-10.pa1[] .f1.w-80.pa1[Cook's Distance]] .pt3[ `$$D_i=\frac{\sum_{j=1}^{n}{(\hat{y}_j-\hat{y}_{j(i)})^2}}{(p+1)\hat{\sigma}^2}$$` - `\(\hat{y}+j\)` is the `\(j\)`th fitted value - `\(\hat{y}_{j(i)}\)` is the `\(j\)`th value from a fit which doesn't include observation `\(i\)` - `\(p\)` is the number of regression coefficients - `\(\hat{\sigma}^2\)` is the estimated variance from the fit, i.e., mean squared error ] --- # Cook's Distance .pull-left[ ```r plot(mod,which=4) ``` - observations labelled by row - various rules of thumb, but start looking when Cook's Distance > 0.5 ] .pull-right[  ] --- class: animated, rollIn # Learning to Read .pull-left[  ] .pull-right[ - the Playmo School has been evaluating its reading programmes, using 50 students - ages of students - hours per week students spend reading of their own volition - whether they are taught using phonics or whole-word methods - **outcome: "reading age"** ] --- count: false # Learning to Read .pull-left[  ] .pull-right[ .center[ <style>html { font-family: -apple-system, 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padding-right: 5px; border-top-style: double; border-top-width: 6px; border-top-color: #D3D3D3; } #chzwqgmpak .gt_striped { background-color: rgba(128, 128, 128, 0.05); } #chzwqgmpak .gt_table_body { border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; } #chzwqgmpak .gt_footnotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #chzwqgmpak .gt_footnote { margin: 0px; font-size: 90%; padding: 4px; } #chzwqgmpak .gt_sourcenotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #chzwqgmpak .gt_sourcenote { font-size: 90%; padding: 4px; } #chzwqgmpak .gt_left { text-align: left; } #chzwqgmpak .gt_center { text-align: center; } #chzwqgmpak .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #chzwqgmpak .gt_font_normal { font-weight: normal; } #chzwqgmpak .gt_font_bold { font-weight: bold; } #chzwqgmpak .gt_font_italic { font-style: italic; } #chzwqgmpak .gt_super { font-size: 65%; } #chzwqgmpak .gt_footnote_marks { font-style: italic; font-size: 65%; } </style> <div id="chzwqgmpak" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table"> <thead class="gt_col_headings"> <tr> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">age</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">hrs_wk</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">method</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">R_AGE</th> </tr> </thead> <tbody class="gt_table_body"> <tr> <td class="gt_row gt_right">10.266</td> <td class="gt_row gt_right">5.821</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right">15.962</td> </tr> <tr> <td class="gt_row gt_right">7.385</td> <td class="gt_row gt_right">4.281</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right">11.598</td> </tr> <tr> <td class="gt_row gt_right">10.445</td> <td class="gt_row gt_right">5.552</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right">15.062</td> </tr> <tr> <td class="gt_row gt_right">6.189</td> <td class="gt_row gt_right">3.063</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right">8.065</td> </tr> <tr> <td class="gt_row gt_right">10.463</td> <td class="gt_row gt_right">5.690</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right">15.125</td> </tr> <tr> <td class="gt_row gt_right">7.801</td> <td class="gt_row gt_right">5.158</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right">7.194</td> </tr> <tr> <td class="gt_row gt_right">9.981</td> <td class="gt_row gt_right">3.487</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right">10.208</td> </tr> <tr> <td class="gt_row gt_right">8.715</td> <td class="gt_row gt_right">5.110</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right">8.268</td> </tr> <tr> <td class="gt_row gt_right">7.408</td> <td class="gt_row gt_right">3.435</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right">6.570</td> </tr> <tr> <td class="gt_row gt_right">7.811</td> <td class="gt_row gt_right">5.625</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right">6.718</td> </tr> </tbody> </table></div> ]] ??? - this is just some of the data in the dataframe - on the right you can see the outcome variable, `R_AGE` for reading age. - note that I tend to put my dependent variables in all caps, just so that I can recognise at a glance that they're what we measured. --- count: false # Learning to Read .pull-left[  ] .pull-right[ .center[ <style>html { font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif; } #jluvseuccd .gt_table { display: table; border-collapse: collapse; margin-left: auto; margin-right: auto; color: #333333; font-size: 16px; font-weight: normal; font-style: normal; background-color: #FFFFFF; width: auto; border-top-style: solid; border-top-width: 2px; border-top-color: #A8A8A8; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #A8A8A8; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; } #jluvseuccd .gt_heading { background-color: #FFFFFF; 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90%; padding: 4px; } #jluvseuccd .gt_sourcenotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #jluvseuccd .gt_sourcenote { font-size: 90%; padding: 4px; } #jluvseuccd .gt_left { text-align: left; } #jluvseuccd .gt_center { text-align: center; } #jluvseuccd .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #jluvseuccd .gt_font_normal { font-weight: normal; } #jluvseuccd .gt_font_bold { font-weight: bold; } #jluvseuccd .gt_font_italic { font-style: italic; } #jluvseuccd .gt_super { font-size: 65%; } #jluvseuccd .gt_footnote_marks { font-style: italic; font-size: 65%; } </style> <div id="jluvseuccd" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table"> <thead class="gt_col_headings"> <tr> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">age</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">hrs_wk</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">method</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">R_AGE</th> </tr> </thead> <tbody class="gt_table_body"> <tr> <td class="gt_row gt_right">10.266</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.821</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.962</td> </tr> <tr> <td class="gt_row gt_right">7.385</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.281</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">11.598</td> </tr> <tr> <td class="gt_row gt_right">10.445</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.552</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.062</td> </tr> <tr> <td class="gt_row gt_right">6.189</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">3.063</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.065</td> </tr> <tr> <td class="gt_row gt_right">10.463</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.690</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.125</td> </tr> <tr> <td class="gt_row gt_right">7.801</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.158</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">7.194</td> </tr> <tr> <td class="gt_row gt_right">9.981</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">3.487</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.208</td> </tr> <tr> <td class="gt_row gt_right">8.715</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.110</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.268</td> </tr> <tr> <td class="gt_row gt_right">7.408</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">3.435</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.570</td> </tr> <tr> <td class="gt_row gt_right">7.811</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.625</td> <td class="gt_row gt_center">word</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.718</td> </tr> </tbody> </table></div> ]] --- # Does Practice Affect Reading Age? .pull-left[ ```r p <- reading %>% ggplot(aes(x=hrs_wk,y=R_AGE)) + geom_point(size=3) + ylab("reading age") + xlab("hours reading/week") p ``` ] .pull-right[  ] -- - hours per week is correlated with reading age: `\(r=0.3483\)`, `\(p=0.0132\)` - we can use a linear model to say something about the effect size --- # Does Practice Affect Reading Age? .pull-left[ ```r p + geom_smooth(method="lm") ``` ] .pull-right[  ] -- - each extra hour spent reading a week adds 1.21 years to reading age --- # A Linear Model ```r mod <- lm(R_AGE ~ hrs_wk, data=reading) summary(mod) ``` ``` ## ## Call: ## lm(formula = R_AGE ~ hrs_wk, data = reading) ## ## Residuals: ## Min 1Q Median 3Q Max *## -5.10 -2.73 0.13 2.42 5.27 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 3.639 1.857 1.96 0.0559 . *## hrs_wk 1.211 0.351 3.46 0.0012 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 3.03 on 48 degrees of freedom *## Multiple R-squared: 0.199, Adjusted R-squared: 0.183 *## F-statistic: 11.9 on 1 and 48 DF, p-value: 0.00116 ``` --- class: center, middle, inverse .f1[but...] --- .center[ <!-- --> ] --- # Assumptions Not Met! .pull-left[ - it seems that the assumptions aren't met for this model - (another demonstration on the right) - one reason for this can be because there's still systematic structure in the residuals - i.e., _more than one thing_ which can explain the variance ] .pull-right[ <!-- --> ] --- class: inverse, center, middle, animated, flip # End of Part 2 --- class: inverse, center, middle # Part 3 ## Multiple Regression --- # Adding Age into the Equation .pull-left[ - so far, have focused on effects of practice - but presumably older children read 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border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #ygnkjrteau .gt_sourcenote { font-size: 90%; padding: 4px; } #ygnkjrteau .gt_left { text-align: left; } #ygnkjrteau .gt_center { text-align: center; } #ygnkjrteau .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #ygnkjrteau .gt_font_normal { font-weight: normal; } #ygnkjrteau .gt_font_bold { font-weight: bold; } #ygnkjrteau .gt_font_italic { font-style: italic; } #ygnkjrteau .gt_super { font-size: 65%; } #ygnkjrteau .gt_footnote_marks { font-style: italic; font-size: 65%; } </style> <div id="ygnkjrteau" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table"> <thead class="gt_col_headings"> <tr> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">age</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">hrs_wk</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">method</th> <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">R_AGE</th> </tr> </thead> <tbody class="gt_table_body"> <tr> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.266</td> <td class="gt_row gt_right">5.821</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.962</td> </tr> <tr> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">7.385</td> <td class="gt_row gt_right">4.281</td> <td class="gt_row gt_center">phonics</td> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">11.598</td> </tr> <tr> <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.445</td> <td class="gt_row gt_right">5.552</td> <td class="gt_row 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] ??? - **rotate cube leftward to show R_AGE~age** --- # Another Model .center[ <!-- --> ] --- # Another Model ```r mod2 <- lm(R_AGE ~ age, data=reading) summary(mod2) ``` ``` ## ## Call: ## lm(formula = R_AGE ~ age, data = reading) ## ## Residuals: ## Min 1Q Median 3Q Max *## -4.097 -2.655 -0.474 2.630 5.685 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 1.867 1.857 1.00 0.32 *## age 1.010 0.228 4.42 0.000056 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 2.86 on 48 degrees of freedom *## Multiple R-squared: 0.289, Adjusted R-squared: 0.275 *## F-statistic: 19.6 on 1 and 48 DF, p-value: 0.0000558 ``` ??? - point out that intercept is quite meaningless (reading age for age zero) --- .center[ <!-- --> ] --- # Two Models, No Answers .pull-left[ - we now have two models that don't map well to assumptions - each suggests an effect + one of `age` + one of `hrs_wk` ] .pull-right[ - if we run them independently, the chances of a type 1 error are + `\(\frac{1}{20}\)` (`mod`, including `hrs_wk`) + `\(\frac{1}{20}\)` (`mod2`, including `age`) - or ** `\(\frac{1}{10}\)` ** overall ] -- .pt1[ &nbsp; ] .br3.pa2.bg-gray.white.tc[ we need to test multiple predictors in _one_ linear model ] --- # Model Equations Again `$$\color{red}{\textrm{outcome}_i}=\color{blue}{(\textrm{model})_i}+\textrm{error}_i$$` `$$\color{red}{y_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_i}+\epsilon_i$$` -- ### linear model with two predictors `$$\color{red}{y_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_{1i}+b_2\cdot{}x_{2i}}+\epsilon_i$$` `$$\color{red}{\hat{y}_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_{1i}+b_2\cdot{}x_{2i}}$$` -- .center[ `y ~ 1 + x1 + x2` `R_AGE ~ 1 + hrs_wk + age` &nbsp;&nbsp; or &nbsp;&nbsp; `R_AGE ~ hrs_wk + age`¹ ] .footnote[ ¹ we'll come back to why order can matter in a bit ] --- # Running A Multiple Regression ```r mod.m <- lm(R_AGE ~ age + hrs_wk, data=reading) summary(mod.m) ``` ``` ## ## Call: ## lm(formula = R_AGE ~ age + hrs_wk, data = reading) ## ## Residuals: ## Min 1Q Median 3Q Max ## -4.249 -2.266 0.742 2.351 3.453 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) -2.092 2.128 -0.98 0.33039 *## age 0.882 0.214 4.12 0.00015 *** *## hrs_wk 0.966 0.309 3.12 0.00307 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 2.63 on 47 degrees of freedom ## Multiple R-squared: 0.412, Adjusted R-squared: 0.386 ## F-statistic: 16.4 on 2 and 47 DF, p-value: 0.00000388 ``` --- # Running a Multiple Regression ``` ## ... ## (Intercept) -2.092 2.128 -0.98 0.33039 ## age 0.882 0.214 4.12 0.00015 *** ## hrs_wk 0.966 0.309 3.12 0.00307 ** ## ... ``` - there are _independent_ effects of age and practice + reading age improves by 0.8817 for each year of age + reading age improves by 0.9661 for each weekly hour of practice - note that the _intercept_ (0 years old, 0 hours/week) is meaningless here -- - important question: is this model _better_ than a model based just on age? --- # Model Fit ``` ## ... ## Residual standard error: 2.63 on 47 degrees of freedom ## Multiple R-squared: 0.412, Adjusted R-squared: 0.386 ## F-statistic: 16.4 on 2 and 47 DF, p-value: 0.00000388 ``` - in multiple regression, `\(R^2\)` measures the fit of the entire model + sum of individual `\(R^2\)`s _if predictors not correlated_ - `\(R^2=0.4115\)` looks better than the `\(R^2\)` for `mod2` (`age` as a predictor) of `\(0.2895\)` - but _any_ predictor will improve `\(R^2\)` (chance associations guarantee this) ```r *mod.r <- update(mod2, ~ . + runif(50)) summary(mod.r) ``` ``` ## ... ## Multiple R-squared: 0.32, Adjusted R-squared: 0.291 ## ... ``` ??? - Adjusted `\(R^2\)` accounts for multiple predictors - for `mod2`, it's 0.2747 - for `mod.r`, it's 0.2909 --- # Comparing Models ``` ## ... ## Multiple R-squared: 0.32, Adjusted R-squared: 0.291 *## F-statistic: 11 on 2 and 47 DF, p-value: 0.000117 ``` - `\(F\)` ratio compares model to the null model, which just has an intercept - but we can also use `\(F\)` ratios to compare models in succession ```r anova(mod.m) ``` ``` ## Analysis of Variance Table ## ## Response: R_AGE ## Df Sum Sq Mean Sq F value Pr(>F) ## age 1 160 159.5 23.12 0.000016 *** ## hrs_wk 1 67 67.3 9.75 0.0031 ** ## Residuals 47 324 6.9 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- # Order Matters! - `age` then `hrs_wk` ``` ## Analysis of Variance Table ## ## Response: R_AGE ## Df Sum Sq Mean Sq F value Pr(>F) ## age 1 160 159.5 23.12 0.000016 *** ## hrs_wk 1 67 67.3 9.75 0.0031 ** ## Residuals 47 324 6.9 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` - `hrs_wk` then `age` ``` ## Analysis of Variance Table ## ## Response: R_AGE ## Df Sum Sq Mean Sq F value Pr(>F) ## hrs_wk 1 110 109.8 15.9 0.00023 *** ## age 1 117 117.0 16.9 0.00015 *** ## Residuals 47 324 6.9 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- # Type 1 vs. Type 3 SS - order matters because R, by default, uses **Type 1** sums of squares + calculate each predictor's improvement to the model _in turn_ - compare to **Type 3** sums of squares + calculate each predictor's improvement to the model _taking all other predictors into account_ - huge debate about which is "better" (nobody likes Type 2) - if using Type 1, _predictors should be entered into the model in a theoretically-motivated order_ --- # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- count: false # Type 1 vs. Type 3 SS .pull-left[ ### type 1 .center[  ]] .pull-right[ ### type 3 .center[  ]] --- # The Two-Predictor Model ```r mod.m <- lm(R_AGE ~ age + hrs_wk, data=reading) summary(mod.m) ``` ``` ## ## Call: ## lm(formula = R_AGE ~ age + hrs_wk, data = reading) ## ## Residuals: ## Min 1Q Median 3Q Max ## -4.249 -2.266 0.742 2.351 3.453 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) -2.092 2.128 -0.98 0.33039 *## age 0.882 0.214 4.12 0.00015 *** *## hrs_wk 0.966 0.309 3.12 0.00307 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 2.63 on 47 degrees of freedom ## Multiple R-squared: 0.412, Adjusted R-squared: 0.386 ## F-statistic: 16.4 on 2 and 47 DF, p-value: 0.00000388 ``` --- # The Two-Predictor Model .flex.items-center[ .w-25[ &nbsp; ] .w-50[ <div id="rgl98101" style="width:504px;height:504px;" class="rglWebGL html-widget"></div> <script type="application/json" 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] .w-25[ &nbsp; ]] --- class: inverse, center, middle, animated, flip # End --- # Acknowledgements - icons by Diego Lavecchia from the [Noun Project](https://thenounproject.com/)