Data set

Today we will work with the prejudice_df object. The data frame contains the following four variables:

• student_id: numeric student identification
• prej_0: prejudice level under \(Y_{0i}\) (Contact=0)
• prej_1: prejudice level under \(Y_{1i}\) (Contact=1)
• dorm_type: binary for actual treatment state

## # A tibble: 8 x 4
##   student_id prej_0 prej_1 dorm_type
##        <dbl>  <dbl>  <dbl>     <dbl>
## 1          1      6      5         0
## 2          2      4      2         1
## 3          3      4      4         0
## 4          4      6      7         0
## 5          5      3      1         1
## 6          6      2      2         1
## 7          7      8      7         0
## 8          8      4      5         0

a) Individual Treatment Effect (ITE)

We assume from the potential outcomes framework that each subject has a potential outcome under both treatment states. Let's take the first student in the list as an example.

The figure illustrates the potential outcomes for Student 1.

We see that in a reality where Student 1 is assigned to in-group dorm (contact=0) their levels of prejudice are 6. On the contrary, in a reality where Student 1 is assigned to co-ethnic dorm (contact=1) their levels of prejudice are 5.

From this illustration, we can gather the individual treatment effect (ITE) for student one. The ITE is equal to the values under treatment (contact=1) minus to the values without treatment (contact=0) or \(ITE = y_{1i} - y_{0i}\).

\[ITE = 5 - 6 = -1\]

Evidently, each subject can only be observed in one treatment state at any point in time in real life. This is known as the fundamental problem (Holland, 1986) of causal inference. The Individual Treatment Effect (ITE) in reality is unattainable. Still, it provides us with a conceptual foundation for causal estimation.

Exercise: Our data are coming from a world with perfect information. In that sense, we have both potential outcomes prej_0 and prej_1. Can you think of a way to calculate the ITE for the eight students with one of the dplyr verbs we learned in the previous section?

#you can employ dplyr::mutate() to create the new variable ite
prejudice_df %>% 
  dplyr::mutate(ite = prej_1 - prej_0)

Average Treatment Effect (ATE)

Normally, we are not interested in the estimates of individual subjects, but rather a population. The Average Treatment Effect (ATE) is the difference in the average potential outcomes of the population.

\[ATE = E(Y_{1i}) - E(Y_{0i})\]

In other words, the ATE is the average ITE of all the subjects in the population. As you can see, the ATE as defined in the formula is also not attainable. Can you think why?

Exercise: Since our data are coming from a world with perfect information. Can you think of a way to calculate the ATE for the eight students based on what we learned last week?

#we know that the ATE is the averge of every subject's ITE. Do you remember dplyr::summarize()?
#how can we use the verbs from last week to get the average treatment effect?

prejudice_df %>%
  dplyr::mutate(ite = prej_1 - prej_0) %>%
  dplyr::summarize(mean(ite))

The Average Treatment Effect Among the Treated and Control (ATT) and (ATC)

The names for these two estimates are very self-explanatory. These two estimates are simply the average treatment effects conditional on the group subjects are assigned to.

The average treatment effect on the treated ATT is defined as the difference in the average potential outcomes for those subjects who were treated: \[ATT = E(Y_{1i}-Y_{0i} | D = 1)\]

The average treatment effect on the control ATC is defined as the difference in the average potential outcomes for those subjects who were not treated: \[ATU = E(Y_{1i}-Y_{0i} | D = 0)\]

Exercise: Since our data are coming from a world with perfect information. Can you think of a way to calculate the ATT and ATC for the eight students based on what we learned last week?

#we know that the ATT and ATU are the average of every subject's ITE grouped by their treatment status. Do you remember how the combination of dplyr::group_by() and dplyr::summarize() worked?
#how can we use the verbs from last week to get the average treatment effect on the treated and untreated?

prejudice_df %>%
  dplyr::mutate(ite = prej_1 - prej_0) %>%
  dplyr::group_by(dorm_type) %>%
  dplyr::summarize(mean(ite))

The Naive Average Treatment Effect (NATE)

So far, we have worked with perfect information. Still, we know that in reality we can only observe subjects in one treatment state. This is the information we do have.

\[NATE = E(Y_{1i}|D{i}=1) - E(Y_{0i}|D{i}=0)\] *reads in English as: "The expected average outcome under treatment for those treated minus the expected average outcome under control for those not treated"

Exercise: Can you think of a way to calculate the NATE for the eight students employing the new observed_prej variable?

prejudice_df %>%
  dplyr::mutate(observed_prej = ifelse(dorm_type == 1, prej_1, prej_0))

#we know that the NATE is the difference in average observed outcomes grouped by their treatment status. Do you remember how the combination of dplyr::group_by() and dplyr::summarize() worked?

prejudice_df %>%
  dplyr::mutate(observed_prej = ifelse(dorm_type == 1, prej_1, prej_0)) %>%
  dplyr::group_by(dorm_type) %>%
  dplyr::summarize(mean(observed_prej))

Note. The ìfelse() function is a very handy tool to have. It allows us to generate conditional statements. The syntax is the following:

ifelse(condition_to_meet, what_to_do_if_met, what_to_do_if_not_met)

In the case of observed_prej, we ask R to create a new variable, where if the subject is in a co-ethnic dorm, we print the prejudice value under treatment. If that condition is not met, we print the prejudice value under control.

Bias

During the lecture, we met two sources of bias:

Selection bias

Selection bias is difference in expected outcomes in the absence of treatment for the actual treatment and control group. In other words, these are the underlying differences that individuals in either group start off with.

Heterogeneous Treatment Effect (HTE) bias

Heterogeneous Treatment Effect (HTE) bias is the difference in returns to treatment (the treatment effect) between the treatment and control group, multiplied by the share of the population in control. In other words, this type of bias relates to the dissimilarities stemming for ways in which individuals in either group are affected differently by the treatment.

We will let you think about these for the mock assignment

Exercise: Since our data are coming from a world with perfect information. Can you think of a way to explore the existence selection bias in our data?

Exercise: Since our data are coming from a world with perfect information. Can you think of a way to explore the existence HTE bias in our data?