Power Analysis

class: center, middle, inverse, title-slide

.title[
# Power Analysis 
]
.subtitle[
## Data Analysis for Psychology in R 2 
]
.author[
### dapR2 Team
]
.institute[
### Department of Psychology The University of Edinburgh
]

---

# `pwr` for F-tests
+ For linear models, we use `pwr.f2.test()`

```r
pwr.f2.test(u = , #numerator degrees of freedom (model)
            v = , #denominator degrees of freedom (residual)
            f2 = , #stat to be calculated (below)
            sig.level = ,
            power = 
            )
```

+ `u` and `v` come from study design.
  + `u` = predictors in the model ( `$k$` ) 
  + `v` = n-k-1

+ There are two versions of `$f^2$`
  + these are specified as formula
  + you can also use a pre-selected value; Cohen suggests f2 values of .02, .15, and .35 reflect small, moderate, and large effect sizes. 
  
  
---
# `pwr` for F-tests
+ The first is:

`$$f^2 = \frac{R^2}{1-R^2}$$`
+ This should be used when we want to see the overall power of a set of predictors
  + Think overall model `$F$`-test

+ For example, if we wanted sample size for an overall `$R^2$` of 0.10, with 5 predictors, power of 0.8 and `$\alpha$` = .05

```r
pwr.f2.test(u = 5, #numerator degrees of freedom (model)
            #v = , #denominator degrees of freedom (residual)
            f2 = 0.10/(1-0.10), #stat to be calculated (below)
            sig.level = .05,
            power = .80
            )
```

---
# `pwr` for F-tests

```
## 
##      Multiple regression power calculation 
## 
##               u = 5
##               v = 115.1043
##              f2 = 0.1111111
##       sig.level = 0.05
##           power = 0.8
```

+ We need a sample of ~121 (115 + 5 + 1)

---
# `pwr` for F-tests
+ The second is:

`$$f^2 = \frac{R^2_{AB} - R^2_{A}}{1-R^2_{AB}}$$`

+ This is the power for the incremental-F or the difference between a restricted ( `$R^2_A$` ) and a full ( `$R^2_{AB}$` ) model.

+ For example, if we wanted sample size for a difference between 0.10 (model with 2 predictors) and 0.15 (model with 5 predictors), power of 0.8 and `$\alpha$` = .05

```r
pwr.f2.test(u = 3, #numerator degrees of freedom (model)
            #v = , #denominator degrees of freedom (residual)
            f2 = (0.15 - 0.10)/(1-0.15), #stat to be calculated (below)
            sig.level = .05,
            power = .80
            )
```

---
# `pwr` for F-tests

```r
pwr.f2.test(u = 3, #numerator degrees of freedom (model)
            #v = , #demoninator degrees of freedom (residual)
            f2 = (0.15 - 0.10)/(1-0.15), #stat to be calculated (below)
            sig.level = .05,
            power = .80
            )
```

```
## 
##      Multiple regression power calculation 
## 
##               u = 3
##               v = 185.2968
##              f2 = 0.05882353
##       sig.level = 0.05
##           power = 0.8
```

+ We need a sample of ~180 (174.4 + 5 + 1)