Sums of Squares

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# Sums of Squares 
## Data Analysis for Psychology in R 2 
### Alex Doumas and Tom Booth
### Department of Psychology The University of Edinburgh
### AY 2020-2021

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# Topics for today
- Types of sums of squares. 
- Examples of partitioning variance with sums of squares.

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# Types of sums of squares
- Recall that in a single predictor model, `$\beta_1$` is the slope of the regression line, or the change in `$y$` due to a unit change in `$x$`. 
- In ANOVA, a main-effect of IV1 indicates a change in the DV for changes in IV1, when each other IVj ( `$i \neq j$` ) are held constant 
- But what does all that mean?
  - It could mean: each IVi main effect tells you the contribution of `$x_i$` beyond the contribution of the other predictors already considered. 
  - It could mean: each IVi main effect tells you the contribution of `$x_i$` above and beyond the contribution of all the other predictors? 
  - Depends on how you calculate your SS...

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# Sums of Squares
+ We have been talking a lot about sums of squares calculations.
  + Something we have held off discussing (until now) is that there are different types.

+ **Type I**: Sequential sums of squares
  + Effect of X on Y, holding all previous X constant
  + Order of variables into the model matters

+ **Type III**: Simultaneous sums of squares
  + Effect on X on Y holding all X constant
  + Order of variables into the model does not matter

+ Note: There are also type II sums of squares

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# Example for SS
+ Let’s say we had three predictors of aggression:
  + Age ( `$x_1$` )
  + Gender ( `$x_2$` )
  + Anger-proneness ( `$x_3$` )

+ In type I sums of squares:
  + The effect of age would be evaluated holding no other variables constant
  + The effect of gender would be evaluated holding age constant
  + The effect of anger-proneness would be evaluated holding age and gender constant

+ In type III sums of squares:
  + The effects of age, gender and anger-proneness would all be evaluated holding every other IV constant

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# SS formally

`$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$$`

.pull-left[
**Type I**

`$$SS(\beta_1)$$`
`$$SS(\beta_2 | \beta_1)$$`
`$$SS(\beta_3 | \beta_1, \beta_2)$$`

]

.pull-right[

**Type III**

`$$SS(\beta_1 | \beta_2, \beta_3)$$`
`$$SS(\beta_2 | \beta_1, \beta_3)$$`
`$$SS(\beta_3 | \beta_1, \beta_2)$$`
]

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# SS Type I (sequential)

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# SS Type I (sequential) x1

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# SS Type I (sequential) x1

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# SS Type I (sequential) x1

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# SS Type I (sequential) x3

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# SS Type I (sequential) x3

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# SS Type I (sequential) x3

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# SS Type I (sequential)
+ Comparison

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# SS Type III (simultaneous)

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# Equivalent SS

+ If **all predictors are uncorrelated** Type I and Type III sums of squares provide the same partition

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# Types of sums of squares: Summary
- Ways to calculate the effects of each predictor. 
  - Type I SS: calculate the improvement to the model produced by each predictor considered in turn. 
  - Type III SS: calculate the improvement to the model produced by each predictor, when taking all of the other predictors into account simultaneously.

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# SS in R
+ `lm` uses Type III sums of squares

+ `anova` uses Type I sums of squares

+ `ezANOVA` (and some other packages) you can specify

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# Summary
- There are different ways to calculate sums of squares. 
- Most packages will use Type I or Type III sums of squares. 
- The main point to consider is whether the sums of squares calculation is sequential or simultaneous.