Partitioning variability1 / 24

Partitioning variability

2 / 24

by Dr. Lucy D'Agostino McGowan

Why?y−¯y=(^y−¯y)+(y−^y)y−y¯=(y^−y¯)+(y−y^)
3 / 24

Why?

$y - \bar{y} = (\hat{y} - \bar{y}) + (y - \hat{y})$
$\sum (y - \bar{y})^{2} = \sum (\hat{y} - \bar{y})^{2} + \sum (y - \hat{y})^{2}$

3 / 24

Why?

$y - \bar{y} = (\hat{y} - \bar{y}) + (y - \hat{y})$
$\sum (y - \bar{y})^{2} = \sum (\hat{y} - \bar{y})^{2} + \sum (y - \hat{y})^{2}$
SSTotal = SSModel + SSE

3 / 24

by Dr. Lucy D'Agostino McGowan

Degrees of freedomSSTotal: n−1n−1
4 / 24

by Dr. Lucy D'Agostino McGowan

Degrees of freedomSSTotal: n−1n−1
SSE: n−2n−2
4 / 24

by Dr. Lucy D'Agostino McGowan

Degrees of freedomSSTotal: n−1n−1
SSE: n−2n−2
SSModel: n−1=1+(n−2)n−1=1+(n−2) - so 1!
4 / 24

by Dr. Lucy D'Agostino McGowan

Mean SquaresMSModel=SSModel1MSModel=SSModel1
5 / 24

by Dr. Lucy D'Agostino McGowan

Mean SquaresMSModel=SSModel1MSModel=SSModel1
MSE=SSEn−2MSE=SSEn−2
5 / 24

$F = \frac{M S M o d e l}{M S E}$

6 / 24

F-distribution

Under the null hypothesis

7 / 24

Sparrows

We can see all of these pieces using the anova() function

lm(Weight ~ WingLength, data = Sparrows) %>%
  anova()

## Analysis of Variance Table
## 
## Response: Weight
##             Df Sum Sq Mean Sq F value    Pr(>F)
## WingLength   1 355.05  355.05  181.25 < 2.2e-16
## Residuals  114 223.31    1.96

8 / 24

by Dr. Lucy D'Agostino McGowan

Sparrowslm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
9 / 24

by Dr. Lucy D'Agostino McGowan

Sparrowslm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
F-statistic: 181.25
9 / 24

by Dr. Lucy D'Agostino McGowan

Sparrowslm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
F-statistic: 181.25
p-value: 2.62e-25 
9 / 24

by Dr. Lucy D'Agostino McGowan

Sparrowslm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
F-statistic: 181.25
p-value: 2.62e-25 
Where did this p-value come from?
9 / 24

p-value

The probability of getting a statistic as extreme or more extreme than the observed test statistic given the null hypothesis is true

10 / 24

F-distribution

Under the null hypothesis

11 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
12 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
SSE: ?
12 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
SSE: ?
SSModel: ?
12 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
13 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
SSE: n−2n−2 = 114
13 / 24

by Dr. Lucy D'Agostino McGowan

Sparrows: Degrees of freedomSSTotal: n−1n−1 = 115
SSE: n−2n−2 = 114
SSModel: 115 - 114 = 1
13 / 24

Sparrows

To calculate the p-value under the t-distribution we used pt(). What do you think we use to calculate the p-value under the F-distribution?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

14 / 24

Sparrows

To calculate the p-value under the t-distribution we used pt(). What do you think we use to calculate the p-value under the F-distribution?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

pf()

15 / 24

Sparrows

To calculate the p-value under the t-distribution we used pt(). What do you think we use to calculate the p-value under the F-distribution?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

pf()
it takes 3 arguments: q, df1, and df2. What do you think df1 and df2 are?

15 / 24

Sparrows

To calculate the p-value under the t-distribution we used pt(). What do you think we use to calculate the p-value under the F-distribution?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

pf(181.2535, 1, 114, lower.tail = FALSE)

## [1] 2.621946e-25

16 / 24

Sparrows

Why don't we multiple this p-value by 2 when we use pf()?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

pf(181.2535, 1, 114, lower.tail = FALSE)

## [1] 2.621946e-25

17 / 24

F-Distribution

Under the null hypothesis

We observed an F-statistic of 181.25, but for demonstration purposes, let's assume we saw one of 2.

18 / 24

F-Distribution

Under the null hypothesis

We observed an F-statistic of 181.25, but for demonstration purposes, let's assume we saw one of 2.

19 / 24

F-Distribution

Under the null hypothesis

Are there any negative values in an F-distribution?

20 / 24

F-Distribution

Under the null hypothesis

The p-value calculates values "as extreme or more extreme", in the t-distribution "more extreme values", defined as farther from 0, can be positive or negative. Not so for the F!

21 / 24

by Dr. Lucy D'Agostino McGowan

SparrowsNotice the p-value for the F-test is the same as the p-value for the t-test for β1β1!lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
lm(Weight ~ WingLength, data = Sparrows) %>% 
  tidy()

## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1
## 2 WingLength     0.467    0.0347     13.5  2.62e-25
22 / 24

Sparrows

What is the F-test testing?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

23 / 24

Sparrows

What is the F-test testing?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

null hypothesis: the fit of the intercept only model (with ${\hat{β}}_{0}$ only) and your model (in this case, ${\hat{β}}_{0} + {\hat{β}}_{1} x$ ) are equivalent

23 / 24

Sparrows

What is the F-test testing?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

null hypothesis: the fit of the intercept only model (with ${\hat{β}}_{0}$ only) and your model (in this case, ${\hat{β}}_{0} + {\hat{β}}_{1} x$ ) are equivalent
alternative hypothesis: The fit of the intercept only model is significantly worse compared to your model

23 / 24

Sparrows

What is the F-test testing?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

null hypothesis: the fit of the intercept only model (with ${\hat{β}}_{0}$ only) and your model (in this case, ${\hat{β}}_{0} + {\hat{β}}_{1} x$ ) are equivalent
alternative hypothesis: The fit of the intercept only model is significantly worse compared to your model
When we only have one variable in our model, $x$ , the p-values from the F and t are going to be equivalent

23 / 24

Sparrows

How are the test statistics related between the F and the t?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

lm(Weight ~ WingLength, data = Sparrows) %>% 
  tidy()

## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1
## 2 WingLength     0.467    0.0347     13.5  2.62e-25

24 / 24

Sparrows

How are the test statistics related between the F and the t?

lm(Weight ~ WingLength, data = Sparrows) %>% 
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

lm(Weight ~ WingLength, data = Sparrows) %>% 
  tidy()

## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1
## 2 WingLength     0.467    0.0347     13.5  2.62e-25

13.5^2

## [1] 182.25

## [1] 182.25

24 / 24

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Partitioning variability

Partitioning variability

Why?

Why?

Why?

SSTotal = SSModel + SSE

Degrees of freedom

SSTotal: n−1n−1

Degrees of freedom

SSTotal: n−1n−1

SSE: n−2n−2

Degrees of freedom

SSTotal: n−1n−1

SSE: n−2n−2

SSModel: n−1=1+(n−2)n−1=1+(n−2) - so 1!

Mean Squares

Mean Squares

F-distribution

Sparrows

Sparrows

Sparrows

Sparrows

Sparrows

p-value

F-distribution

Sparrows: Degrees of freedom

Sparrows: Degrees of freedom

Sparrows: Degrees of freedom

Sparrows: Degrees of freedom

Sparrows: Degrees of freedom

Sparrows: Degrees of freedom

Sparrows

Sparrows

Sparrows

Sparrows

Sparrows

F-Distribution

F-Distribution

F-Distribution

F-Distribution

Sparrows

Notice the p-value for the F-test is the same as the p-value for the t-test for β1β1!

Sparrows

Sparrows

Sparrows

Sparrows

Sparrows

Sparrows

Partitioning variability

Help

SSTotal: $n - 1$

SSTotal: $n - 1$

SSE: $n - 2$

SSTotal: $n - 1$

SSE: $n - 2$

SSModel: $n - 1 = 1 + (n - 2)$ - so 1!

Notice the p-value for the F-test is the same as the p-value for the t-test for $β_{1}$ !