class: center, middle, inverse, title-slide # <b>Week 7: The Linear Model</b> ## Univariate Statistics and Methodology using R ### Martin Corley ### Department of Psychology<br/>The University of Edinburgh --- class: inverse, center, middle # Part 1: Correlation++ --- # Back on the Road .pull-left[ <!-- --> .center[ `\(r= 0.4648\)`, `\(p = 0.0007\)` ] ] .pull-right[  ] ??? - is there anything more the police' data about blood alcohol and reaction time can tell us? - so far we know they're correlated, but it would be good if we could something about _how bad_ blood alcohol is + how much worse does drinking more make things? --- # Back on the Road (2) .pull-left[ <!-- --> .tc[ "for every extra 0.01% blood alcohol, reaction time slows down by around 32 ms" ] ] .pull-right[  ] ??? - this kind of information is based on the assumption that the relationship between blood alcohol and RT is _linear_ - that is, that each additional unit of blood alcohol affects reaction time by the same amount --- # The Only Equation You Will Ever Need .br3.pa2.f2.white.bg-gray[ $$ \textrm{outcome}_i = (\textrm{model})_i + \textrm{error}_i $$ ] .center.pt2[  ] --- count: false # The Only Equation You Will Ever Need .br3.pa2.f2.white.bg-gray[ $$ \textrm{outcome}_i = (\textrm{model})_i + \textrm{error}_i $$ ] - to get any further, we need to make _assumptions_ - nature of the **model** .tr[ (linear) ] - nature of the **errors** .tr[ (normal) ] --- # A Linear Model .flex.items-top[ .w-40.pa2[ $$ \color{red}{\textrm{outcome}_i} = \color{blue}{(\textrm{model})_i} + \textrm{error}_i $$ $$ \color{red}{y_i} = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} + \epsilon_i $$ .center[ so the linear model itself is... ] $$ \hat{y}_i = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} $$ .center[  ] ] .w-60[ &nbsp; ]] --- count: false # A Linear Model .flex.items-top[ .w-40.pa2[ $$ \color{red}{\textrm{outcome}_i} = \color{blue}{(\textrm{model})_i} + \textrm{error}_i $$ $$ \color{red}{y_i} = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} + \epsilon_i $$ .center[ so the linear model itself is... ] $$ \hat{y}_i = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} $$ .center[  ]] .w-60.pa2[ <!-- --> ]] --- count: false # A Linear Model .flex.items-top[ .w-40.pa2[ $$ \color{red}{\textrm{outcome}_i} = \color{blue}{(\textrm{model})_i} + \textrm{error}_i $$ $$ \color{red}{y_i} = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} + \epsilon_i $$ .center[ so the linear model itself is... ] $$ \hat{y}_i = \color{blue}{b_0 \cdot{} 1 + b_1 \cdot{} x_i} $$ .center[  ] .br3.pa3.bg-light-yellow[ $$ \hat{y} = \color{blue}{b_0 + b_1 \cdot{} x_i} $$ .center[  ]] ] .w-60.pa2[ <!-- --> ]] --- # Take An Observation .flex.items-top[ .w-40.pa2[ .br3.bg-gray.white.f2.pa2.tc[ x_i = 1.2, y_i = 9.9 ] $$ \hat{y}_i = b_0 + b_1\cdot{}x_i = 7.4 $$ $$ y_i = \hat{y}_i + \epsilon_i = 7.4 + \color{red}{2.5} $$ ] .w-60.pa2[ <!-- --> ] ] ??? - `\(\hat{y}_i\)` is what the model _predicts_ for `\(x_i\)` - `\(y_i\)` is the actual value that was observed for `\(x_i\)` - why would we care? + for one thing, the model can predict `\(\hat{y}\)` for values of `\(x\)` that we have never observed --- # Simplifying the Data .pull-left[ - **NB** we're doing this for illustrative purposes only ```r ourDat <- dat %>% sample_n(20) # take 20 data points ``` ] .pull-right[ <!-- --> ] --- count: false # Simplifying the Data .pull-left[ - **NB** we're doing this for illustrative purposes only ```r ourDat <- dat %>% sample_n(20) # take 20 data points ``` ```r ourDat <- ourDat %>% mutate(BloodAlc = BloodAlc*100) ``` ] .pull-right[ <!-- --> ] --- count: false # Simplifying the Data .pull-left[ - **NB** we're doing this for illustrative purposes only ```r ourDat <- dat %>% sample_n(20) # take 20 data points ``` ```r ourDat <- ourDat %>% mutate(BloodAlc = BloodAlc*100) ``` .center[ "for every extra unit blood alcohol, reaction time slows down by around 33 ms" ] ] .pull-right[ <!-- --> ] -- .center.f3.pt3[ but how can we evaluate our model? ] --- # Linear Models in R ```r mod <- lm(RT ~ BloodAlc, data=ourDat) ``` --- count: false # Linear Models in R ```r mod <- lm(RT ~ BloodAlc, data=ourDat) summary(mod) ``` ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = ourDat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -121.18 -39.10 -4.71 44.55 121.14 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 322.6 158.0 2.04 0.056 . *## BloodAlc 32.7 15.4 2.12 0.048 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 61.7 on 18 degrees of freedom ## Multiple R-squared: 0.2, Adjusted R-squared: 0.156 ## F-statistic: 4.51 on 1 and 18 DF, p-value: 0.0479 ``` --- # Intercept and Slope Again $$ b_0= 322.6; \quad b_1= 32.7 $$ .center[ <!-- --> ] ??? - `\(b_0\)` is the the predicted value, `\(\bar{y}\)`, when `\(x=0\)`. Here I've represented it as a line, so where the top is the intercept value of 322.629 --- count: false # Intercept and Slope Again $$ b_0= 322.6; \quad b_1= 32.7 $$ .center[ <!-- --> ] --- count: false # Intercept and Slope Again $$ b_0= 322.6; \quad b_1= 32.7 $$ .center[ <!-- --> ] --- count: false # Intercept and Slope Again $$ b_0= 322.6; \quad b_1= 32.7 $$ .center[ <!-- --> ] ??? - note that the intercept is really very far from the data we're interested in. - it may be pretty meaningless to talk about the reaction time of someone with zero blood alcohol --- class: inverse, center, middle, animated, fadeInDownBig # End of Part 1 --- class: inverse, center, middle # Part 2 ## Significance --- # Intercept and Slope ```r mod <- lm(RT ~ BloodAlc, data=ourDat) summary(mod) ``` ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = ourDat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -121.18 -39.10 -4.71 44.55 121.14 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 322.6 158.0 2.04 0.056 . *## BloodAlc 32.7 15.4 2.12 0.048 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 61.7 on 18 degrees of freedom ## Multiple R-squared: 0.2, Adjusted R-squared: 0.156 ## F-statistic: 4.51 on 1 and 18 DF, p-value: 0.0479 ``` --- # Are We Impressed? - we have an intercept of 322.6 and a slope of 32.7 - in NHST world, our pressing question is -- .pt4[ &nbsp; ] .br3.bg-gray.pa2.white.tc[ how likely would we have been to find these parameters under the null hypothesis? ] --- # Testing Chance .center[ <!-- --> ] - repeatedly sampling 20~datapoints from the population + variability in _height_ of line = variability in intercept ( `\(b_0\)` ) + variability in _angle_ of line = variability in slope ( `\(b_1\)` ) --- # We've Seen This Before .center[ <!-- --> ] - shaded area represents "95% confidence interval" + if we repeatedly sampled 20 items from the population... + assumes that the 20 we have are the _best estimate_ of the population ??? - if you were able to lay these graphs over each other, they wouldn't be the same - in the simulation graph, we _know_ the entire population (of 50 drink-drivers) we're repeatedly sampling 20 from - in this graph, the only information we have is that our 20 drink-drivers represent "the population" but we have no other information about that population + so the 20 are our "best estimate" of the population + **just like means and standard errors** (here the edges of the grey regions are 2.1009 standard errors from the line) --- # The Good Old _t_-Test ``` ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 322.6 158.0 2.04 0.056 . ## BloodAlc 32.7 15.4 2.12 0.048 * ``` - for each model parameter we are interested in whether it is _different from zero_ - **intercept**: just like a mean - **slope**: does the best-fit line differ from horizontal? -- - these are just (two-tailed) one-sample `\(t\)`-tests + **standard error** is the standard deviation of doing these lots of times + **t value** is `\(\frac{\textrm{Estimate}}{\textrm{Std. Error}}\)` + to calculate `\(p\)`, we need to know the _degrees of freedom_ --- # Degrees of Freedom .center[ <!-- --> ] --- count: false # Degrees of Freedom .center[ <!-- --> ] ??? - we can always add in two points to make the straight line that we want to see - this is one way of showing that for `\(n\)` data points, there are `\(n-2\)` degrees of freedom --- # Degrees of Freedom - in fact we subtract 2 degrees of freedom because we "know" two things + intercept ( `\(b_0\)` ) + slope ( `\(b_1\)` ) - the remaining degrees of freedom are the _residual_ degrees of freedom -- - the _model_ also has associated degrees of freedom + 2 (intercept, slope) - 1 (knowing one affects the other) .br3.bg-gray.pa2.white.tc[ the models we have been looking at have 20 observations and 1 predictor **(1, 18)** degrees of freedom ] --- # Linear Models in R ```r mod <- lm(RT ~ BloodAlc, data=ourDat) summary(mod) ``` ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = ourDat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -121.18 -39.10 -4.71 44.55 121.14 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 322.6 158.0 2.04 0.056 . ## BloodAlc 32.7 15.4 2.12 0.048 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 61.7 on 18 degrees of freedom *## Multiple R-squared: 0.2, Adjusted R-squared: 0.156 *## F-statistic: 4.51 on 1 and 18 DF, p-value: 0.0479 ``` --- # Total Sum of Squares $$ \textrm{total SS}=\sum{(y - \bar{y})^2} $$ .pull-left[ - sum of squares between observed `\(y\)` and mean `\(\bar{y}\)` - represents the total amount of variance in the model - how much does the observed data vary from a model which says "there is no effect of `\(x\)`" (**null model**)? ] .pull-right[ <!-- --> ] --- # Residual Sum of Squares $$ \textrm{residual SS} = \sum{(y - \hat{y})^2} $$ .pull-left[ - sum of squared differences between observed `\(y\)` and predicted `\(\hat{y}\)` - represents the unexplained variance in the model - how much does the observed data vary from the existing model? ] .pull-right[ <!-- --> ] --- # Model Sum of Squares $$ \textrm{model SS} = \sum{(\hat{y} - \bar{y})^2} $$ .pull-left[ - sum of squared differences between predicted `\(\hat{y}\)` and mean `\(\bar{y}\)` - represents the additional variance explained by the current model over the null model ] .pull-right[ <!-- --> ] --- # Testing the Model: _R_² .pull-left[ $$ R^2 = \frac{\textrm{model SS}}{\textrm{total SS}} = \frac{\sum{(\hat{y}-\bar{y})^2}}{\sum{(y-\bar{y})^2}} $$ "how much the model improves over the null" .pt3[ - `\(0 \le R^2 \le 1\)` - we want `\(R^2\)` to be large - for a single predictor, `\(\sqrt{R^2} = |r|\)` ]] .pull-right[ <!-- --> ] --- # Testing the Model: _F_ .pull-left[ `\(F\)` ratio depends on **mean squares** <br/> .tc[ ( `\(\textrm{MS}_x = \textrm{SS}_x/\textrm{df}_x\)` ) ] `$$F=\frac{\textrm{model MS}}{\textrm{residual MS}}=\frac{\sum{(\hat{y}-\bar{y})^2}/\textrm{df}_m}{\sum{(y-\hat{y})^2}/\textrm{df}_r}$$` "how much the model improves over chance" .pt3[ - `\(0 < F\)` - we want `\(F\)` to be large - significance of `\(F\)` does not always equate to a large (or theoretically sensible) effect ]] .pull-right[ <!-- --> ] --- # A Linear Model for 20 Drinkers .center[ <!-- --> ] - a linear model describes the **best-fit line** through the data - minimises the error terms `\(\epsilon\)` or **residuals** --- # Two Types of Significance ```r mod <- lm(RT ~ BloodAlc, data=ourDat) summary(mod) ``` ``` ## ## Call: ## lm(formula = RT ~ BloodAlc, data = ourDat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -121.18 -39.10 -4.71 44.55 121.14 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) *## (Intercept) 322.6 158.0 2.04 0.056 . *## BloodAlc 32.7 15.4 2.12 0.048 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 61.7 on 18 degrees of freedom *## Multiple R-squared: 0.2, Adjusted R-squared: 0.156 *## F-statistic: 4.51 on 1 and 18 DF, p-value: 0.0479 ``` --- # The Good, the Bad, and the Ugly .flex.items-center[ .w-50.br3.bg-washed-green.pa2[ - we can easily extend this approach + use more than one predictor + generalised linear model ] .w-50.br3.bg-washed-red.pa2[ - not a panacea + depends on _assumptions_ about the data + depends on _decisions_ about analysis ]] -- .flex.items-center[ .w-70.br3.pa2.bg-gray.white[ - like other statistics, linear models don't tell you "about" your data - they simply assess what is (un)likely to be due to chance - the key to good statistics is _common sense and good interpretation_ ] .w-30.center[  ] ] --- class: inverse, center, middle, animated, fadeInDownBig # End