Regression Example

Lion data Table 17.1-1, pg 542

Regression of Lion Data

Fit Regression models

## Null model: no relationship between how black nose is and true age
lion.null <- lm(age.years ~ 1,   # What does "~ 1 " tell R?
               data = lions)


## ALt model: Age can be predicted by portion of nose that is black
lion.alt <- lm(age.years ~ portion.black, 
               data = lions)

Visualize the hypotheses posed by the models

Can you draw the hypotheses?

Regression Model Output

Standard way of getting output with summary() command

## 
## Call:
## lm(formula = age.years ~ portion.black, data = lions)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.5457 -1.1457 -0.3384  0.9245  4.3426 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     0.9262     0.5591   1.657    0.108    
## portion.black  10.5827     1.4884   7.110 6.59e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.66 on 30 degrees of freedom
## Multiple R-squared:  0.6276, Adjusted R-squared:  0.6152 
## F-statistic: 50.55 on 1 and 30 DF,  p-value: 6.59e-08

This gives a lot of somewhat cluttered output

Just look at regression coefficients

##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)      0.926      0.559   1.657    0.108
## portion.black   10.583      1.488   7.110    0.000

Questions you should be able to answer:

• Why is one of the p-values equal to zero?
• Which one is the slope of the line?
• What is the biological meaning of a non-significant intercept?

The regression model tells us the equation of the line

##   (Intercept) portion.black 
##     0.9262292    10.5826652

##   (Intercept) portion.black 
##     0.9262292    10.5826652

Predictions from a model

• Say we take a picture of a lion and determine that the portion of its nose that is blck is 0.65 (65%)

• What is the predicted age of the lion

##   (Intercept) portion.black 
##     0.9262292    10.5826652

Points along the regression lines can be thought of as predictions for each value of x if we found a new lion with the value of x.

##   (Intercept) portion.black 
##     0.9262292    10.5826652

##   (Intercept) portion.black 
##     0.9262292    10.5826652

##   (Intercept) portion.black 
##     0.9262292    10.5826652

##   (Intercept) portion.black 
##     0.9262292    10.5826652

Getting predictions in R

• Uses predict() function
• I have written a function called easy.predict() to help with this

easy.predict <- function(model,datum){
  x.var.name <- names(model$coefficients)[[2]]
new.df <- data.frame(datum)
names(new.df) <- x.var.name

  predict(model, newdata = new.df)[[1]]
}

Get prediction for portion blakc = 0.65

easy.predict(lion.alt, datum = 0.65)

## [1] 7.804962

Plot the prediciton

Is regression appropriate for these data

Do we violate the assumptions of the regression model?

Regression assumptions

Shorthand versions (You MUST know these from memory) * 1)Random sampling * 2)Variability around regression line is approximately normal * 3)Variance in y does not change as x-changes

Checking Assumption: Model diagnostics

• This can be thought of as “Analysis of the residuals”
• residuals = (real data) - (predictions from model)

You get the residuals in R with the resid() command

#Get the residuals
model.residuals <- resid(lion.alt)

#Look at first 12
model.residuals[1:12]

##           1           2           3           4           5           6 
## -2.04858891 -0.90780235 -0.19032239 -0.10197570  0.40385095  0.89802430 
##           7           8           9          10          11          12 
##  1.00385095  0.06889104 -0.96024222 -1.15441556 -1.14276226 -0.08449574

Note that they can be both positive AND negative. Why is that?

Assumption 1: random sampling

Frequently violated, especially with regression…

Assumption 2: normality

The most basic model diagnostic: are residuals normal-ish distributed

use the hist() command

hist(model.residuals)

• Kinda look skewed…
• …but not horrible
• Normality is NOT a particularly important assumption

Assumption 2: Normality

Better look at normality: qqplot in R

“qq” = “quantile quantile”

plot(lion.alt, which = 2)

3: Constant variance

plot(lion.alt, which = 1)