1.1 Absolute Criteria

Bias characterizes whether an estimator tends to lie above or below the true parameter, on average. Variance characterizes the precision of an estimator, as the average squared deviation of the estimator from its average. Note that variance is the inverse of precision. Therefore, the higher the variance, the lower the precision. Standard error, also characterizing precision, is the square root of variance. Mean squared error (MSE) and root mean squared error (RMSE) characterize the accuracy of the estimates. MSE and RMSE measure how far off, on average, an estimator is from the true parameter. Absolute criteria are in the same scale as the estimate. MSE is in squared deviation scale and RMSE is in the scale of the estimates.

Let \(T\) denote an estimator for a parameter \(\theta\). By running a simulation study, we obtain \(K\) estimates \(T_1,...,T_K\) (or realizations of the estimator) for each data generating condition. We can calculate the following sample statistics for the estimates:

• Sample mean: \(\bar{T} = \frac{1}{K}\sum_{k=1}^K T_k\)
• Sample variance: \(S_T^2 = \frac{1}{K - 1}\sum_{k=1}^K \left(T_k - \bar{T}\right)^2\)
• Sample skewness (standardized): \(g_T = \frac{1}{K S_T^3}\sum_{k=1}^K \left(T_k - \bar{T}\right)^3\)
• Sample kurtosis (standardized): \(k_T = \frac{1}{K S_T^4} \sum_{k=1}^K \left(T_k - \bar{T}\right)^4\)

Table 1 below shows each of the performance criteria, its interpretation, its formal definition, how the criterion is estimated in a simulation study, and the formula for the MCSE of the estimated performance measure.

The equation for the MCSE of the standard deviation is derived using the jack-knife technique (Efron & Stein, 1981). First we calculate variance of the estimates leaving out replicate \(j\). Instead of calculating each jack-knife estimate, we use algebraic tricks to calculate \(S^2_{T(j)}\) as follows:

\[S_{T(j)}^2 = \frac{1}{K - 2} \left[(K - 1) S_T^2 - \frac{K}{K - 1}\left(T_j - \bar{T}\right)^2\right]\] Then, the jack-knife MCSE of standard deviation will be calculated as:

\[\sqrt{\frac{K - 1}{K} \sum_{j=1}^K (\sqrt{S_{T(j)}^2} - S_T)^2 }\]

The equation for the MCSE of RMSE is also derived using the jack-knife technique, which involves excluding a replicate \(j\) and calculating RMSE (Efron & Stein, 1981). The formula for RMSE is:

\[\sqrt{\frac{1}{K}\sum_{k=1}^{K}\left(T_k - \theta\right)^2}\]

This is approximately equivalent to:

\[RMSE = \sqrt{(\bar{T} - \theta)^2 + S^2_T}\]

The jack-knife RMSE will be calculated as:

\[RMSE_{(j)} = \sqrt{(\bar{T}_{(j)} - \theta)^2 + S^2_{T(j)}}\]

Here \(\bar{T}_{(j)}\) and \(S^2_{T(j)}\) indicate the mean and variance of the estimates leaving out replicate \(j\). Instead of calculating each jack-knife estimate, we use algebraic tricks to calculate \(\bar{T}_{(j)}\) and \(S^2_{T(j)}\) as follows:

\[\bar{T}_{(j)} = \frac{1}{K-1} \left(K \bar{T} - T_j\right)\]

\[S_{T(j)}^2 = \frac{1}{K - 2} \left[(K - 1) S_T^2 - \frac{K}{K - 1}\left(T_j - \bar{T}\right)^2\right]\]

Then, the jack-knife MCSE of RMSE will be calculated as:

\[MCSE_{RMSE(JK)} = \sqrt{\frac{K -1}{K}\sum_{j=1}^K \left(RMSE_{(j)} - RMSE\right)^2}\]

library(simhelpers)
library(dplyr)
library(tibble)
library(knitr)
library(dplyr)
library(kableExtra)

welch_res %>%
  glimpse()
#> Rows: 16,000
#> Columns: 11
#> $ n1          <dbl> 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50…
#> $ n2          <dbl> 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50…
#> $ mean_diff   <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
#> $ iterations  <dbl> 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000…
#> $ seed        <dbl> 204809087, 204809087, 204809087, 204809087, 204809087, 204…
#> $ method      <chr> "t-test", "Welch t-test", "t-test", "Welch t-test", "t-tes…
#> $ est         <dbl> 0.025836000, 0.025836000, 0.005158587, 0.005158587, -0.079…
#> $ var         <dbl> 0.09543914, 0.09543914, 0.08481717, 0.08481717, 0.08179330…
#> $ p_val       <dbl> 0.9335212, 0.9335804, 0.9859039, 0.9859109, 0.7807543, 0.7…
#> $ lower_bound <dbl> -0.5872300, -0.5899041, -0.5727856, -0.5741984, -0.6473703…
#> $ upper_bound <dbl> 0.6389020, 0.6415761, 0.5831027, 0.5845155, 0.4877263, 0.4…

# using do()
welch_res %>%
  filter(method == "t-test") %>% # filter just conventional t-test res
  group_by(n1, n2, mean_diff) %>% # grouping 
  do(calc_absolute(., estimates = est, true_param = mean_diff)) %>% # run the function
  kable(digits = 5) # create a kable table 

# using group_modify()
welch_res %>%
  filter(method == "t-test") %>% # filter just conventional t-test res
  mutate(params = mean_diff) %>% # group_modify cannot take in a group column as an argument
  group_by(n1, n2, mean_diff) %>% # grouping 
  group_modify(~ calc_absolute(.x, estimates = est, true_param = params)) %>%
  kable(digits = 5)

1.2 Relative Criteria

Relative criteria can be useful for describing an estimator’s performance, especially if the performance varies in proportion to the true value of the target parameter. It can be only used when \(|\theta| > 0\) as we cannot divide by \(0\) (Morris et al., 2019).

To derive the MCSE for relative RMSE, we again used the jack-knife technique. The formula to calculate relative RMSE is:

\[rRMSE = \sqrt{\frac{(\bar{T} - \theta)^2 + S_T^2}{\theta^2}}\]

The jack-knife RMSE will be calculated as:

\[rRMSE_{(j)} = \sqrt{\frac{(\bar{T}_{(j)} - \theta)^2 + S_{T(j)}^2}{\theta^2}}\]

Here \(\bar{T}_{(j)}\) and \(S^2_{T(j)}\) are calculated as specified above when we described the algebra trick to estimate these values. The MCSE is then calculated as specified in the table below.

Table 2 below shows each of the relative performance criteria, its interpretation, its formal definition, how the criterion is estimated in a simulation study, and its MCSE formula.

# using group_modify()
welch_res %>%
  filter(method == "t-test") %>%
  mutate(params = mean_diff) %>%
  group_by(n1, n2, mean_diff) %>%
  group_modify(~ calc_relative(.x, estimates = est, true_param = params)) %>%
  kable(digits = 5)

1.3 Relative Criteria for Variance Estimators

Variance estimators are always positive, and so relative criteria are often used to characterize their performance. For variance estimators, we have \(V\) denoting the sampling variance of a point estimator \(T\). To assess the relative criteria for \(V\), we need to divide it by the true value of the sampling variance of \(T\), \(\lambda = \text{Var}(T)\), which we may not be able to calculate directly. In such scenario, we can use the sample variance of \(T\) across the replications, \(S_T^2\), to estimate the true sampling variance.

The relative bias would then be estimated by \(rB = \bar{V} / S_T^2\), the average of the variance estimates divided by the sample variance of the point estimates. To estimate MCSE of the relative bias of the variance estimator, we need to account for the uncertainty in the estimation of the true sampling variance. One way to do so is to use the jack-knife technique that we described above, which entails excluding a replicate \(j\) and calculating relative bias \(\bar{V}_{(j)}/ S_{T(j)}^2\). The Monte Carlo standard error can then be calculated as:

\[ MCSE\left(rB\right) = \sqrt{\frac{K - 1}{K} \sum_{j=1}^K \left(rB_{(j)} - rB\right)^2} \] which can be written as:

\[ MCSE\left(rB\right) = \sqrt{\frac{K - 1}{K} \sum_{j=1}^K \left(\frac{\bar{V}_{(j)}}{S_{T(j)}^2} - \frac{\bar{V}}{S_T^2}\right)^2} \]

We reformulate the MCSE using some algebra tricks similar to how we reformulated them for the RMSE MCSE formulas above.

\[ \begin{aligned} \bar{V}_{(j)} &= \frac{1}{K - 1}\left(K \bar{V} - V_j\right) \\ S_{T(j)}^2 &= \frac{1}{K - 2} \left[(K - 1) S_T^2 - \frac{K}{K - 1}\left(T_j - \bar{T}\right)^2\right] \end{aligned} \]

Similarly, we can estimate the MCSE of relative MSE and RMSE using the jack-knife technique. To estimate the MSE we need to estimate \(S_{V(j)}^2\), which represents the sample variance of the variance estimates leaving replicate \(j\) out. We calculate \(S_{V(j)}^2\) as

\[S_{V(j)}^2 = \frac{1}{K - 2} \left[(K - 1) S_V^2 - \frac{K}{K - 1}\left(V_j - \bar{V}\right)^2\right].\]

We can then estimate the jack-knife relative MSE and RMSE following the same logic as we described above and then calculate the MCSE.

Table 3 below lists relative performance measures for variance estimators.

The function calc_relative_var() calculates the relative performance criteria estimates and corresponding jack-knife MCSEs for a variance estimator. The function requires res_dat, a data frame or tibble containing simulation results, estimates, the name of the column containing point estimates, var_estimates, the name of the column containing the variance estimates, and perfm_criteria, the criteria to be evaluated: perfm_criteria = c("relative bias", "relative mse", "relative rmse").

Below we demonstrate the use of the calc_relative_var() function. The variance estimates are expected to be different when derived from the conventional t-test as opposed to when they are derived from the Welch t-test. We group by the sample sizes, mean difference, and the t-test method. Note the difference between the conventional t-test and the Welch t-test results, especially when the group sample sizes are unequal.

If there are any convergence issues with estimation, please again make sure that the estimation function returns NA values for the variance estimate and the point estimate. The calc_relative_var() function will omit the pair of point estimate and variance estimate if either value is NA before calculating the performance measure and MCSE.

welch_res %>%
  group_by(n1, n2, mean_diff, method) %>%
  group_modify(~ calc_relative_var(.x, estimates = est, var_estimates = var)) %>%
  kable(digits = 5)

1.4 Hypothesis Testing and Confidence Intervals

When doing hypothesis tests, we are often interested in whether the Type 1 error rate is adequately controlled and whether the test has enough power to detect an effect size of substantive interest. The rejection rate of a hypothesis test captures the proportion of times the p-value is below a specified \(\alpha\) level—that is, the proportion of times we reject the null hypothesis. When the specified effect size is 0, we can examine Type 1 error rates and when the magnitude of the effect is greater than 0, we can examine power. We are also interested in confidence interval coverage, the proportion of intervals that contain the true parameter, and the interval width, which is an indicator of the precision of the interval estimator.

Table 4 below presents the performance criteria used to evaluate hypothesis tests. In the table, let \(P_k\) denote the p-value from simulation replication \(k\), for \(k = 1,...,K\). Suppose that the confidence intervals are for the target parameter \(\theta\) and have coverage level \(\beta\). Let \(A_k\) and \(B_k\) denote the lower and upper end-points of the confidence interval from simulation replication \(k\), and let \(W_k = B_k − A_k\), for \(k = 1,...,K\).

# using group_modify()
welch_res %>%
  group_by(n1, n2, mean_diff, method) %>%
  group_modify(~ calc_rejection(.x, p_values = p_val)) %>%
  kable(digits = 5)

# using group_modify()
welch_res %>%
  mutate(params = mean_diff) %>%
  group_by(n1, n2, mean_diff, method) %>%
  group_modify(~ calc_coverage(.x, lower_bound = lower_bound, upper_bound = upper_bound, true_param = params)) %>%
  kable(digits = 5)