[1] '2.1.5'With the arguments theta = TRUE,
full_Output = TRUE and family = "gaussian",
the output will automatically contain the \(\beta\) vector and the \(R\) matrix (i.e., beta_gen
will be called automatically from within
questionnaire_gen).
We generate one latent trait, two continuous, one binary, and one
3-category covariates. The data is generated from a multivariate normal
distribution. The logical argument full_output is
TRUE.
set.seed(1234)
bg <- questionnaire_gen(n_obs = 1000, n_X = 2, n_W = list(2, 3), theta = TRUE, family = "gaussian",
full_output = TRUE)'data.frame': 1000 obs. of 6 variables:
$ subject: int 1 2 3 4 5 6 7 8 9 10 ...
$ theta : num -1.732 0.707 0.911 1.509 -0.5 ...
$ q1 : num -0.491 0.16 -0.39 -1.307 0.602 ...
$ q2 : num 0.5499 0.0669 0.087 -1.628 0.2559 ...
$ q3 : Factor w/ 2 levels "1","2": 2 2 2 1 2 2 1 1 2 1 ...
$ q4 : Factor w/ 3 levels "1","2","3": 1 2 3 2 1 2 1 1 1 3 ...linear_regression is a list that contains two elements.
The first element, betas, summarizes the true regression
coefficients \(\beta\). The second
element, vcov_YXW, shows the \(R\) matrix.
$betas
theta q1 q2 q3.2 q4.2 q4.3
-0.8174844 -0.5836818 -0.5402292 -0.1049199 0.8699444 1.7398887
$vcov_YXW
theta q1 q2 q3.2 q4.2 q4.3
theta 1.00000000 -0.17564245 -0.26487195 -7.889541e-02 0.000000e+00 0.12421718
q1 -0.17564245 1.00000000 -0.28027150 3.541595e-02 0.000000e+00 0.14963273
q2 -0.26487195 -0.28027150 1.00000000 2.556079e-01 0.000000e+00 0.07965237
q3.2 -0.07889541 0.03541595 0.25560791 2.500000e-01 -2.775558e-17 0.06097692
q4.2 0.00000000 0.00000000 0.00000000 -2.775558e-17 2.400000e-01 -0.12000000
q4.3 0.12421718 0.14963273 0.07965237 6.097692e-02 -1.200000e-01 0.21000000beta_gen uses the output from
questionnaire_gen to generate linear regression
coefficients.
theta x1 x2 w12 w22 w23
-0.8174844 -0.5836818 -0.5402292 -0.1049199 0.8699444 1.7398887 If the logical argument MC is TRUE in
beta_gen, Monte Carlo simulation is used to estimate
regression coefficients. If the logical argument
rename_to_q is TRUE, the background variables
are all labeled as q to match the default behavior of
questionnaire_gen.
The first column contains the true \(\beta\), as estimated by the covariance
matrix, which will always be the same for the same data. The column of
MC reports the Monte Carlo simulation estimates for \(\beta\), which is sample-dependent and will
change each time the beta_gen function is called. The next
two columns summarize the 99% confidence interval for these estimates.
And the column of cov_in_CI return to logical argument
whether the cov_matrix estimates are contained within their
respective confidence intervals (“1” corresponds to “yes” and “0” to
“no”).
cov_matrix MC 0.5% 99.5% cov_in_CI
theta -0.8174844 -0.7528447 -0.8380719 -0.64147366 1
q1 -0.5836818 -0.5627695 -0.6279581 -0.49666666 1
q2 -0.5402292 -0.4950472 -0.5571855 -0.44193911 1
q3.2 -0.1049199 -0.1876017 -0.3357658 -0.08231601 1
q4.2 0.8699444 0.8165117 0.7057719 0.92812704 1
q4.3 1.7398887 1.7389447 1.5867220 1.88949655 1 cov_matrix MC 0.5% 99.5% cov_in_CI
theta -0.8174844 -0.7485083 -0.8550206 -0.63323259 1
q1 -0.5836818 -0.5650178 -0.6309527 -0.51627209 1
q2 -0.5402292 -0.4954573 -0.5544919 -0.43396311 1
q3.2 -0.1049199 -0.1793640 -0.3039317 -0.05262306 1
q4.2 0.8699444 0.8092411 0.6481412 0.95344844 1
q4.3 1.7398887 1.7205485 1.5569970 1.90153362 1