class: center, middle, inverse, title-slide # Day 8 ## Linear model - diagnostics ### Michael W. Kearney📊<br />School of Journalism <br />Informatics Institute <br />University of Missouri ### <table style="border-style:none;padding-top:30px;" class=".table"> <tr> <th style="padding-right:75px!important"> <a href="https://twitter.com/kearneymw"> <i class="fab fa-twitter"></i> </a> </th> <th style="padding-left:75px!important"> <a href="https://github.com/mkearney"> <i class="fab fa-github"></i> </a> </th> </tr> <tr style="background-color:#fff"> <th style="padding-right:75px!important"> <a href="https://twitter.com/kearneymw"> <span class="citation">@kearneymw</span> </a> </th> <th style="padding-left:75px!important"> <a href="https://github.com/mkearney"> <span class="citation">@mkearney</span> </a> </th> </tr> </table> --- class: inverse, center, middle ## Linear Model - Pt 3 --- ## Agenda + ANOVA + Interactions + Assumptions --- class: inverse, center, middle ## ANOVA --- ## Calculating ANOVA ANOVA compares variation **between** groups to variation **within** groups + F-statistic is the **ratio** of **between to within** variation + **ANOVA table** simply breaks down the different sources of variation --- ## Sum of Squares - Between + Sum of squared distances between **group means** and **grand mean** (mean differences) `$$SS_{between} = \sum (\bar{x}_{group} - \bar{x}_{grand})^2$$` > To what extent do the _group _means differ from the _grand mean_? --- ## Sum of Squares - Within + Sum of squared distances between **observed values** and **group means** (group variances) `$$SS_{within} = \sum (x_{obser} - \bar{x}_{group})^2$$` > To what extent do the _observed scores_ differ from the _group means_? --- ## Sum of Squares - Total + Sum of squared distances between **observed values** and **grand mean** (total variance) `$$SS_{total} = \sum (x_{obser} - \bar{x}_{grand})^2$$` > To what extent do the _observed scores_ differ from the _grand mean_? --- ## Sum of Squares - Total + The total sum of squares is equal to the sum of the *between* and *within* sum of squares `$$SS_{total} = SS_{between} + SS_{within}$$` > The variation _within groups_ plus the variation _between groups_ --- class: testtable ## Sources of variation | Source | Calculation | |--------------------|----------------------------------------------| | `\(SS_{between}\)` | `\(\sum (\bar{x}_{group} - \bar{x}_{grand})^2\)` | | `\(SS_{within}\)` | `\(\sum (x_{obser} - \bar{x}_{group})^2\)` | | `\(SS_{total}\)` | `\(\sum (x_{obser} - \bar{x}_{grand})^2\)` | --- class: testtable ## Sources of variation | Source | Calculation | |--------------------|----------------------------------------------| | `\(SS_{between}\)` | `\(\sum (\bar{x}_{group} - \bar{x}_{grand})^2\)` | | `\(SS_{within}\)` | `\(\sum (x_{obser} - \bar{x}_{group})^2\)` | | `\(SS_{total}\)` | `\(SS_{between} + SS_{within}\)` | <style> .testtable table {font-size: 140% !important} </style> --- ## ANOVA table | Source | SS | df | MS | F | |---------|----------------------------------------------|----------|-------------------|-------------------| | Between | `\(\sum (\bar{x}_{group} - \bar{x}_{grand})^2\)` | `\(k - 1\)` | `\(SS_{b} / df_{b}\)` | `\(MS_{b} / MS_{w}\)` | | Within | `\(\sum (x_{obser} - \bar{x}_{group})^2\)` | `\(N - k\)` | `\(SS_{w} / df_{w}\)` | | | Total | `\(SS_{b} + SS_{w}\)` | `\(N - 1\)` | | | --- ## ANOVA table | Source | df | Sum Sq | Mean Sq | F value | Pr(>F) | |--------------------|---------|----------|----------|-------------------|--------| | Factor | `\(k\)` | `\(SS_{b}\)` | `\(MS_{b}\)` | `\(MS_{b} / MS_{w}\)` | `\(p\)` | | Residuals | `\(N - k\)` | `\(SS_{w}\)` | `\(MS_{w}\)` | | &nbsp; | --- ## F-statistic + F is a ratio of variances + If ratio is large, the group means are different from each other for reasons other than random chance + If two groups, F maps perfectly onto t-statistic --- ## Bonferroni ``` r pairwise.t.test(data$Value, data$Factor, p.adjust.method = "bonferroni") ``` --- ## Tukey's ``` r TukeyHSD(aov(Value ~ Factor, data), "Factor") ``` --- ## Residuals View the residuals with by accessing `$residuals` from the model object ``` r m$residuals ``` --- ## Checking assumptions? + What kind of plot of the residuals would be helpful? + How can we check for multi-collinearity?