Topics

• Log transformation on the response

• Log transformation on the predictor

Respiratory Rate vs. Age

• A high respiratory rate can potentially indicate a respiratory infection in children. In order to determine what indicates a "high" rate, we first want to understand the relationship between a child's age and their respiratory rate.

• The data contain the respiratory rate for 618 children ages 15 days to 3 years.

• Variables:

  • Age: age in months
  • Rate: respiratory rate (breaths per minute)

Rate vs. Age

Rate vs. Age

Need to transform Y

• Typically, a "fan-shaped" residual plot indicates the need for a transformation of the response variable y
  • log(Y) is the most straightforward to interpret

• When building a model:
  • Choose a transformation and build the model on the transformed data
  • Reassess the residual plots
  • If the residuals plots did not sufficiently improve, try a new transformation!

Log transformation on Y

• If we apply a log transformation to the response variable, we want to estimate the parameters for the model...

  ^log(Y)=ˆβ0+ˆβ1X

• We want to interpret the model in terms of y not log(Y), so we write all interpretations in terms of

Mean and logs

Suppose we have a set of values

x <- c(3, 5, 6, 8, 10, 14, 19)

log_x <- log(x)
mean(log_x)

## [1] 2.066476

xbar <- mean(x)
log(xbar)

## [1] 2.228477

Median and logs

x <- c(3, 5, 6, 8, 10, 14, 19)

log_x <- log(x)
median(log_x)

## [1] 2.079442

median_x <- median(x)
log(median_x)

## [1] 2.079442

Mean, Median, and log

¯log(x)≠log(ˉx)

mean(log_x) == log(xbar)

## [1] FALSE

Median(log(x))=log(Median(x))

median(log_x) == log(median_x)

## [1] TRUE

Mean and median of log(Y)

• Recall that y=β0+β1xi is the mean value of y at the given value xi. This doesn't hold when we log-transform y

• The mean of the logged values is not equal to the log of the mean value. Therefore at a given value of x

Mean and median of log(y)

• However, the median of the logged values is equal to the log of the median value. Therefore,

• If the distribution of log(y) is symmetric about the regression line, for a given value xi,

  Median(log(y))=Mean(log(y))

Interpretation with log-transformed y

• Given the previous facts, if ^log(Y)=ˆβ0+ˆβ1x, then

  Median(ˆY)=exp{ˆβ0}exp{ˆβ1x}

• Intercept: When X=0, the median of Y is expected to be exp{ˆβ0}
  

• Slope: For every one unit increase in X, the median of Y is expected to multiply by a factor of exp{ˆβ1}

log(Rate) vs. Age

log(Rate) vs. Age

log(Rate) vs. Age

Intercept: The median respiratory rate for a new born child is expected to be 46.759 (exp{3.845}) breaths per minute.

Slope: For each additional month in a child's age, the respiratory rate is expected to multiply by a factor of 0.981 (exp{-0.019}).

Confidence interval for βj

• The confidence interval for the coefficient of X describing its relationship with log(Y) is

ˆβj±t∗SE(^βj)

• The confidence interval for the coefficient of x describing its relationship with Y is

Coefficient of Age

We are 95% confident that for each additional month in age, the respiratory rate will multiply by a factor of 0.98 to 0.982 (exp{-0.02} to exp{-0.018}).

Log Transformation on X

Try a transformation on X if the scatterplot shows some curvature but the variance is constant for all values of X

Model with Transformation on X

• Intercept: When log(X)=0, (X=1), Y is expected to be ˆβ0 (i.e. the mean of y is ˆβ0)

• Slope: When X is multiplied by a factor of C, the mean of Y is expected to change by ˆβ1log(C) units

  • Example: when X is multiplied by a factor of 2, y is expected to change by ˆβ1log(2) units

Rate vs. log(Age)

Rate vs. log(Age)

Intercept: The expected (mean) respiratory rate for children who are 1 month old (log(1) = 0) is 50.135 breaths per minute.

Slope: If a child's age doubles, we expect their respiratory rate to decrease by 4.146 (-5.982*log(2)) breaths per minute.

Recap

• Log transformation on the response

• Log transformation on the predictor