Quantitative Research Methods – Introduction to Applied Statistics

Fifth Meeting

• Before-and-after design
• Difference-in-Difference estimate
• Observational studies:
  • Statistics for one variable
  • Quantiles
  • Root of mean squares (RMS)
  • Standard deviation

• Longitudinal / panel data yield more credible results on comparisons between treatment and control groups
• The before-and-after design examines how the outcome variable changed from the pretreatment period to the posttreatment period for the same set of units. The design is able to adjust for any confounding factor that is specific to each unit but does not change over time. However, the design does not address possible bias due to time-varying confounders.

## Full-time employees in the starting period
minwageNJ$fullPropBefore <- minwageNJ$fullBefore / (minwageNJ$fullBefore + minwageNJ$partBefore)

## Difference-in-means
NJdiff <- mean(minwageNJ$fullPropAfter) - mean(minwageNJ$fullPropBefore)
NJdiff

• Difference-in-Difference (Diff-in-Diff) design extends the the before-and-after design and addresses the confounding bias;
• Here we assume that the outcome variable follows a parallel trend when there is no treatment

• In Diff-in-Diff designs, our quantity of interest is the Sample Average Treatment effect for the Treated (SATT)

## Penn: difference in means
minwagePA$fullPropBefore <- minwagePA$fullBefore / (minwagePA$fullBefore + minwagePA$partBefore)


PAdiff <- mean(minwagePA$fullPropAfter) - mean(minwagePA$fullPropBefore)


## NJ: difference in means
minwageNJ$fullPropBefore <- minwageNJ$fullBefore / (minwageNJ$fullBefore + minwageNJ$partBefore)

NJdiff <- mean(minwageNJ$fullPropAfter) - mean(minwageNJ$fullPropBefore)

## Diff-in-diff
NJdiff - PAdiff

• Median describes central value of a distribution. It represents the middle value of a distribution, if the number of observations is odd OR the average of the two values, if the distribution is even
• 
• Medians may sometimes give a useful and more truthful idea about the distribution than mean estimates, imagine calculating an average household income in a country which has several billionaires and many poor people (sound familiar?)

## Median difference between states
median(minwageNJ$fullPropAfter) - median(minwagePA$fullPropAfter)
## Difference-in-medians between states
NJdiff.med <- median(minwageNJ$fullPropAfter) - median(minwageNJ$fullPropBefore)
NJdiff.med
## Median difference-in-difference
PAdiff.med <- median(minwagePA$fullPropAfter) - median(minwagePA$fullPropBefore)
NJdiff.med - PAdiff.med

• In order to describe what different parts of the distribution look like, we use quintiles
  • IQR (interquartile range): difference between 3rd and 1st quartiles

summary(minwageNJ$wageBefore)

summary(minwageNJ$wageAfter)

IQR(minwageNJ$wageBefore)

IQR(minwageNJ$wageAfter)

quantile(minwageNJ$wageBefore, probs = seq(from = 0, to = 1, by = 0.1))

• Another measure of describing the spread of variable is Root Mean Square (RMS)

sqrt(mean((minwageNJ$fullPropAfter - minwageNJ$fullPropBefore)^2))
mean(minwageNJ$fullPropAfter - minwageNJ$fullPropBefore)

• Using root mean square measure, we can describe the dispertion of values from the mean
• Standard deviation is a root mean square from average value of a distribution
• Variance is a square of standard deviation

sd(minwageNJ$fullPropBefore)
sd(minwageNJ$fullPropAfter)
var(minwageNJ$fullPropBefore)
var(minwageNJ$fullPropAfter)