Tutorials
We first call the ciccr package.
To illustrate the usefulness of the package, we use the dataset ACS that is included the package. This dataset is a extract from American Community Survey (ACS) 2018, restricted to white males residing in California with at least a bachelor’s degree. The ACS is an ongoing annual survey by the US Census Bureau that provides key information about US population. We use the following variables:
The binary outcome `Top Income’ (\(Y\)) is defined to be one if a respondent’s annual total pre-tax wage and salary income is top-coded. In our sample extract, the top-coded income bracket has median income $565,000 and the next highest income that is not top-coded is $327,000.
The binary treatment (\(T\)) is defined to be one if a respondent has a master’s degree, a professional degree, or a doctoral degree.
The covariate (\(X\)) is age in years and is restricted to be between 25 and 70.
The original ACS sample is not a case-control sample but we construct one by the following procedure.
- The case sample \((Y=1)\) is composed of 921 individuals whose income is top-coded.
- The control sample \((Y=0)\) of equal size is randomly drawn without replacement from the pool of individuals whose income is not top-coded.
We now construct cubic b-spline terms with three inner knots using the age variable.
Define \(\beta(y) = E [\log \text{OR}(X) | Y = y]\) for \(y = 0,1\), where \(\text{OR}(x)\) is the odds ratio conditional on \(X=x\): \[ \text{OR}(x) = \frac{P(T=1|Y=1,X=x)}{P(T=0|Y=1,X=x)}\frac{P(T=0|Y=0,X=x)}{P(T=1|Y=0,X=x)}. \] Using the retrospective sieve logistic regression model, we estimate \(\beta(1)\) by
results_case = avg_retro_logit(y, t, x, 'case')
results_case$est
#> y
#> 0.7286012
results_case$se
#> y
#> 0.1013445Here, option 'case' refers to conditioning on \(Y=1\).
Similarly, we estimate \(\beta(0)\) by
results_control = avg_retro_logit(y, t, x, 'control')
results_control$est
#> y
#> 0.5469094
results_control$se
#> y
#> 0.1518441Here, option 'control' refers to conditioning on \(Y=1\).
We carry out causal inference by
Here, 0.2 is the specified upper bound for unknown \(p = \text{Pr}(Y=1)\). If it is not specified, the default choice for the upper bound for \(p\) is p_upper = 1. Here, 0.95 refers to the level of the confidence interval (0.95 is the default choice).
The S3 object results contains a grid of estimates est, standard errors se, and one-sided confidence intervals ci ranging from p = 0 to p = p_upper. In addition, the grid pseq from 0 to p_upper is saved as part of the S3 object results.
# point estimates
results$est
#> [1] 0.5469094 0.5488219 0.5507345 0.5526470 0.5545596 0.5564721 0.5583847
#> [8] 0.5602972 0.5622097 0.5641223 0.5660348 0.5679474 0.5698599 0.5717725
#> [15] 0.5736850 0.5755976 0.5775101 0.5794227 0.5813352 0.5832477
# standard errors
results$se
#> [1] 0.1518441 0.1502495 0.1486627 0.1470838 0.1455132 0.1439511 0.1423978
#> [8] 0.1408536 0.1393188 0.1377937 0.1362787 0.1347740 0.1332800 0.1317971
#> [15] 0.1303256 0.1288660 0.1274187 0.1259841 0.1245626 0.1231546
# confidence intervals
results$ci
#> [1] 0.7966706 0.7959604 0.7952628 0.7945784 0.7939075 0.7932506 0.7926083
#> [8] 0.7919808 0.7913688 0.7907728 0.7901933 0.7896308 0.7890860 0.7885594
#> [15] 0.7880516 0.7875633 0.7870953 0.7866480 0.7862224 0.7858191
# grid points from 0 to p_upper
results$pseq
#> [1] 0.00000000 0.01052632 0.02105263 0.03157895 0.04210526 0.05263158
#> [7] 0.06315789 0.07368421 0.08421053 0.09473684 0.10526316 0.11578947
#> [13] 0.12631579 0.13684211 0.14736842 0.15789474 0.16842105 0.17894737
#> [19] 0.18947368 0.20000000To be more compatible with the odds ratio, it is useful to transform them by the exponential function:
# point estimate
exp(results$est)
#> [1] 1.727904 1.731212 1.734527 1.737847 1.741174 1.744507 1.747847 1.751193
#> [9] 1.754545 1.757904 1.761269 1.764641 1.768019 1.771404 1.774795 1.778193
#> [17] 1.781597 1.785008 1.788425 1.791848
# confidence interval estimate
exp(results$ci)
#> [1] 2.218144 2.216569 2.215023 2.213507 2.212023 2.210571 2.209151 2.207765
#> [9] 2.206415 2.205100 2.203822 2.202583 2.201383 2.200224 2.199108 2.198034
#> [17] 2.197005 2.196023 2.195089 2.194203It is handy to examine the results by plotting a graph.
yaxis_limit = c(min(exp(results$est)),(max(exp(results$ci))+0.25*(max(exp(results$ci))-min(exp(results$est)))))
plot(results$pseq, exp(results$est), type = "l", lty = "solid", col = "blue", xlab = "Pr(Y*=1)",ylab = "exp(change in log probability)", xlim = c(0,max(results$pseq)), ylim = yaxis_limit)
lines(results$pseq, exp(results$ci), type = "l", lty = "dashed", col = "red")
To interpret the results, we assume both marginal treatment response (MTR) and marginal treatment selection (MTS). In this setting, MTR means that everyone will not earn less by obtaining a degree higher than bachelor’s degree; MTS indicates that those who selected into higher education have higher potential to earn top incomes. Based on the MTR and MTS assumptions, we can conclude that the treatment effect lies in between 1 and the upper end point of the one-sided confidence interval with high probability. Thus, the estimates in the graph above suggest that the effect of obtaining a degree higher than bachelor’s degree is anywhere between [1, 2.2], which roughly implies that the chance of earning top incomes may increase up to by a factor of around 2, but allowing for possibility of no positive effect at all. In other words, it is unlikely that the probability of earning top incomes will more than double by pursuing higher education beyond BA. See Jun and Lee, 2020 for more detailed explanations regarding how to interpret the estimation results.