# Usage of functions LF and QF in Linear/Logistic regression settings

We propose a few examples on the usage of SIHR to simulated dataset. We will show how to conduct inference for linear functionals (LF) and quadratic functionals (QF) on linear and logistic regression settings, respectively.

library(SIHR)

## Linear Regression Setting

We consider the setting that $$n=200, p=150$$ with $X_i \sim N(\textbf{0}_p, \textbf{I}_p),\; Y_i = \alpha + X_i^\intercal \beta + \epsilon_i, \; \epsilon_i\sim N(0,1),\; \textrm{where }\; \alpha = -0.5, \; \beta = (0.5, \textbf{1}_4, \textbf{0}_{p-5}).$ Our goal is to construct valid inference for objectives:

1. $$\beta_1 = 0.5$$
2. $$\beta_1 + \beta_2 = 1.5$$
3. $$\beta_{G}^\intercal \Sigma_{G,G} \beta_{G} = 3.25$$, where $$\Sigma=\mathbb{E}[X_i^\intercal X_i] = \textbf{I}_p$$ and $$G=\{1,2,3,4\}$$.

The 1st and 2nd objectives will be achieved togther by LF( ), while the 3d objective will be conducted with QF( ).

Generate Data

set.seed(0)
n <- 200
p <- 150
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- -0.5 + X %*% c(0.5, rep(1, 4), rep(0, p - 5))

### LF: Linear Functionals

loading1 <- c(1, rep(0, p - 1)) # for 1st objective, true value = 0.5
loading2 <- c(1, 1, rep(0, p - 2)) # for 2nd objective, true value = 1.5
loading.mat <- cbind(loading1, loading2)

Conduct Inference, call LF with model="linear":

Est <- LF(X, y, loading.mat, model = "linear", intercept = TRUE, intercept.loading = FALSE, verbose = TRUE)
#> The projection direction is identified at mu = 0.044356at step =4
#> The projection direction is identified at mu = 0.044356at step =4

The parameter intercept indicates whether we fit the model with/without intercept term. The parameter intercept.loading indicates whether we include intercept term in the inference objective. In this example, the model is fitted with intercept, but we do not include it in our final objective.

Methods for LF

ci(Est)
#> 1       1 0.4892515 0.5115561
#> 2       2 1.4886438 1.5184631
summary(Est)
#> Call:
#> Inference for Linear Functional
#>
#> Estimators:
#>        1     0.4764     0.5004   0.005690   87.94        0 ***
#>        2     1.4533     1.5036   0.007607  197.65        0 ***

Notice that the true values are $$0.5$$ and $$1.5$$ for 1st and 2nd objective respectively, both are included in their corresponding confidence interval. Also it is evident that our bias-corrected estimators is much closer to the true values than the Lasso estimators.

For quadratic functionals, we need to specify the subset $$G \subseteq [p]$$. If argument $$A$$ is not specified (default = NULL), we will automatically conduct inference on $$\beta_G \Sigma_{G,G} \beta_G$$.

G <- c(1:4) # 3rd objective, true value = 3.25

Conduct Inference, call QF with model="linear". The argument split indicates whether we split samples or not for computing the initial estimator.

Est <- QF(X, y, G, A = NULL, model = "linear", intercept = TRUE, verbose = TRUE)
#> The projection direction is identified at mu = 0.062729at step =3

ci method for QF

ci(Est)
#>    tau    lower    upper
#> 1 0.25 2.239521 3.890422
#> 2 0.50 2.233725 3.896219
#> 3 1.00 2.222250 3.907693

summary method for QF

summary(Est)
#> Call:
#>
#>   tau est.plugin est.debias Std. Error z value  Pr(>|z|)
#>  0.25        2.9      3.065     0.4212   7.278 3.400e-13 ***
#>  0.50        2.9      3.065     0.4241   7.227 4.947e-13 ***
#>  1.00        2.9      3.065     0.4300   7.128 1.016e-12 ***

In the output results, each row represents the result for different values of $$\tau$$, the enlargement factor for asymptotic variance to handle super-efficiency. Notice that the true value is $$3.25$$ for 3rd objective, which is included in the confidence interval.

## Logistic Regression Setting

The procedures of usage in the logistic regression setting are almost the same as the one in linear setting, except that we need to specify the argument model="logistic" or model="logistic_alter", instead of model="linear". We propose two different debiasing methods for logistic regression, both work theoretically and empiricially.

We consider the setting that $$n=200, p=150$$ with $X_i \sim N(\textbf{0}_p, \textbf{I}_p),\; P_i = \frac{\exp(\alpha + X_i^\intercal \beta)}{1+\exp(\alpha + X_i^\intercal \beta)},\; Y_i = {\rm Binomial}(P_i),\; \textrm{where }\; \alpha = -0.5, \;\beta = (0.5, 1, \textbf{0}_{p-2}).$ Our goal is to construct valid inference for objectives:

1. $$\beta_1 + \beta_2 = 1.5$$
2. $$-\frac{1}{2}\beta_1 - \beta_2 = -1.25$$
3. $$\beta_{G}^\intercal \Sigma_{G,G} \beta_{G} = 1.25$$, where $$\Sigma=\mathbb{E}[X_i^\intercal X_i] = \textbf{I}_p$$ and $$G=\{1,2,3\}$$.

The 1st and 2nd objectives will be achieved togther by LF( ), while the 3d objective will be conducted with QF( ).

Generate Data

set.seed(1)
n <- 200
p <- 120
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
val <- -1.5 + X[, 1] * 0.5 + X[, 2] * 1
prob <- exp(val) / (1 + exp(val))
y <- rbinom(n, 1, prob)

### LF: Linear Functionals

loading1 <- c(1, 1, rep(0, p - 2)) # for 1st objective, true value = 1.5
loading2 <- c(-0.5, -1, rep(0, p - 2)) # for 2nd objective, true value = -1.25
loading.mat <- cbind(loading1, loading2)

Conduct Inference, call LF with model="logistic" or model="logistic_alter":

Est <- LF(X, y, loading.mat, model = "logistic", verbose = TRUE)
#> The projection direction is identified at mu = 0.028911at step =5
#> The projection direction is identified at mu = 0.028911at step =5

Methods for LF

ci(Est)
#> 1       1  0.6510009  1.866141
#> 2       2 -1.3927844 -0.424308
summary(Est)
#> Call:
#> Inference for Linear Functional
#>
#> Estimators:
#>        1     0.3745     1.2586     0.3100   4.060 4.907e-05 ***
#>        2    -0.2762    -0.9085     0.2471  -3.677 2.357e-04 ***

Notice that the true values are $$1.5$$ and $$-1.25$$ for 1st and 2nd objective respectively, both are included in their corresponding confidence interval. Also it is evident that our bias-corrected estimators is much closer to the true values than the Lasso estimators.

For quadratic functionals, we find that sufficient larger sample size is needed for better empirical result, since we need to split samples to obtain initial estimators. Thus, we generate another simulated data but with larger sample size $$n=400$$.

set.seed(0)
n <- 400
p <- 120
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
val <- -1.5 + X[, 1] * 0.5 + X[, 2] * 1
prob <- exp(val) / (1 + exp(val))
y <- rbinom(n, 1, prob)
G <- c(1:3) # 3rd objective, true value = 1.25

Conduct Inference, call QF with model="logistic_alter".

Est <- QF(X, y, G, A = NULL, model = "logistic_alter", intercept = TRUE, verbose = TRUE)
#> The projection direction is identified at mu = 0.029056at step =5

ci method for QF

ci(Est)
#>    tau     lower    upper
#> 1 0.25 0.2274503 2.048520
#> 2 0.50 0.1339998 2.141970
#> 3 1.00 0.0000000 2.306665

summary method for QF

summary(Est)
#> Call:
#>  1.00     0.6434      1.138     0.5963   1.908  0.05633 .
In the output results, each row represents the result for different values of $$\tau$$, the enlargement factor for asymptotic variance to handle super-efficiency. Notice that the true value is $$3.25$$ for 3rd objective, which is included in the confidence interval.