Correlated test statistics for

The weighted parametric group sequential design (WPGSD) (Anderson et al. (2022)) approach allows one to take advantage of the known correlation structure in constructing efficacy bounds to control family-wise error rate (FWER) for a group sequential design. Here correlation may be due to common observations in nested populations, due to common observations in overlapping populations, or due to common observations in the control arm.

Methodologies to calculate correlations

Suppose that in a group sequential trial there are $m$ elementary null hypotheses $H_i$ , $i \in I={1,...,m},$ and there are $K$ analyses. Let $k=1,2,...K$ be the index for the interim analyses and final analyses. For any nonempty set $J \subseteq I$ , we denote the intersection hypothesis $H_J=\cap_{j \in J}H_j$ . We note that $H_I$ is the global null hypothesis.

We assume the plan is for all hypotheses to be tested at each of the $k$ planned analyses if the trial continues to the end for all hypotheses. We further assume that the distribution of the $m \times K$ tests of $m$ individual hypotheses at all $k$ analyses is multivariate normal with a completely known correlation matrix.

Let $Z_{ik}$ be the standardized normal test statistic for hypothesis $i \in I$ , analysis $1 \le k \le K$ . Let $n_{ik}$ be the number of events collected cumulatively through stage $k$ for hypothesis $i$ . Then $n_{i \wedge i',k \wedge k'}$ is the number of events included in both $Z_{ik}$ and $i$ , $i' \in I$ , $1 \le k$ , $k' \le K$ . The key for the parametric tests is to utilize the correlation among the test statistics. The correlation between $Z_{ik}$ and $Z_{i'k'}$ is $Corr(Z_{ik},Z_{i'k'})=\frac{n_{i \wedge i',k \wedge k'}}{\sqrt{n_{ik}\times n_{i'k'}}}$ .

Examples

We borrow an Example 1 from Anderson et al. (2022) here. Assume a two-arm controlled clinical trial with one primary endpoint and three patient populations defined by the status of two biomarkers, A and B:

Biomarker A positive (population 1),
Biomarker B positive (population 2),
Overall population (population 3).

The 3 primary elementary null hypotheses are:

H1: the experimental treatment is no different than control in population 1
H2: the experimental treatment is no different than control in population 2
H3: the experimental treatment is no different than control in the overall population

Assume an interim analysis and a final analysis are planned for the study. The number of events observed at each analysis in each population as well as the intersections of populations 1 and 2 to are assumed to be

library(dplyr)
library(tibble)
library(gt)

event_tb <- tribble(
  ~Population, ~"Number of Event at IA", ~"Number of Event at FA",
  "Population 1", 100, 200,
  "Population 2", 110, 220,
  "Overlap of Population 1 and 2", 80, 160,
  "Overall Population", 225, 450
)
event_tb %>%
  gt() %>%
  tab_header(title = "Number of events in each population")

Population	Number of Event at IA	Number of Event at FA
Number of events in each population
Population 1	100	200
Population 2	110	220
Overlap of Population 1 and 2	80	160
Overall Population	225	450

Correlation of tests for different populations at the same analysis

Let’s consider the situation at the interim analysis. The correlation between tests of H1 and H2 will be $Corr(Z_{11},Z_{21})=\frac{n_{1 \wedge 2,1 \wedge 1}}{\sqrt{n_{11}\times n_{21}}}=\frac{80}{\sqrt{100\times 110}}=0.76.$

The 80 in the numerator is the overlap number of events of population 1 and population 2 at the interim analysis.

Correlation for tests of the same population at different analyses

We consider the correlation between tests for population 1 at the IA and FA. The correlation will thus be: $Corr(Z_{11},Z_{12})=\frac{n_{1 \wedge 1,1 \wedge 2}}{\sqrt{n_{11}\times n_{12}}}=100/\sqrt{100\times 200}=0.71.$

Correlation of tests at different analyses for different populations

Next we consider the correlation between the tests of population 1 at the interim analysis and population 2 at the final analyses. The correlation will be $\text{Corr}(Z_{11},Z_{22})=\frac{n_{1 \wedge 1,2 \wedge 2}}{\sqrt{n_{11}\times n_{22}}}=\frac{80}{\sqrt{100\times 220}}=0.54.$ The 80 in the numerator is the overlap number of events of population 1 in interim analysis and population 2 in final analysis; i.e., the events in both populations 1 and 2 at the interim analysis.

Generate the correlation matrix by `s7_generate_corr()`

Now we know how to calculate the correlation values under different situations, and the s7_generate_corr() function was built based on this logic. We can directly calculate the complete correlation matrix for all tests via the function.

First, we need an event table including the event counts for testing individual hypotheses and pairwise intersection hypotheses for the study.

H1, H2 are each the index of a hypothesis. We will have the index for H1 be $<=$ the index for H2 to avoid duplication of pairs.
Analysis indicates which of sequentially indexed analyses, interim and final, is being referred to.
Event is the common number of events observed for testing both H1 and H2 at the specified Analysis.

For example: H1=1, H2=1, Analysis=1, Event=100indicates that in the first population, there are 100 events for comparing the experimental treatment to control at the interim analysis. Thus, when H1=H2, we are looking at the number of events used for testing the simple hypothesis specified in H1=H2.

For testing an intersection hypothesis with H1 != H2 and a given Analysis value, we are looking at the number of events that are used in testing both H1 and H2. For our example above, if H1=1, H2=2, Analysis=2, Event=160 indicates that the number of overlapping cases for comparing the experimental treatment to the control in population 1 and 2 which is 160.

The column names needed for input to s7_generate_corr() are H1, H2, Analysis, Event. We take the event counts from above accordingly.

library(wpgsd)
# The event table
event <- tibble::tribble(
  ~H1, ~H2, ~Analysis, ~Event,
  1, 1, 1, 100,
  2, 2, 1, 110,
  3, 3, 1, 225,
  1, 2, 1, 80,
  1, 3, 1, 100,
  2, 3, 1, 110,
  1, 1, 2, 200,
  2, 2, 2, 220,
  3, 3, 2, 450,
  1, 2, 2, 160,
  1, 3, 2, 200,
  2, 3, 2, 220
)

By specifying this as the S7 type EventTable we get input checks to ensure no missing values or inconsistencies.

event <- as_event_table(event)
event

## <wpgsd::EventTable> function (data = tibble::tibble())  
##  @ data        : tibble [12 × 4] (S3: tbl_df/tbl/data.frame)
##  $ H1      : num [1:12] 1 2 3 1 1 2 1 2 3 1 ...
##  $ H2      : num [1:12] 1 2 3 2 3 3 1 2 3 2 ...
##  $ Analysis: num [1:12] 1 1 1 1 1 1 2 2 2 2 ...
##  $ Event   : num [1:12] 100 110 225 80 100 110 200 220 450 160 ...
##  @ n_hypotheses: int 3
##  @ n_analyses  : int 2

For human readability of the data table:

event@data

## # A tibble: 12 × 4
##       H1    H2 Analysis Event
##    <dbl> <dbl>    <dbl> <dbl>
##  1     1     1        1   100
##  2     2     2        1   110
##  3     3     3        1   225
##  4     1     2        1    80
##  5     1     3        1   100
##  6     2     3        1   110
##  7     1     1        2   200
##  8     2     2        2   220
##  9     3     3        2   450
## 10     1     2        2   160
## 11     1     3        2   200
## 12     2     3        2   220

Now we input the above event table to the function of generate_corr_s7(), and get the correlation matrix as follow.

corr_mat <- generate_corr_s7(event)
corr_mat

## <wpgsd::CorrelationMatrix>
##  @ matrix      : num [1:6, 1:6] 1 0.763 0.667 0.707 0.539 ...
##  .. - attr(*, "dimnames")=List of 2
##  ..  ..$ : chr [1:6] "H1_A1" "H2_A1" "H3_A1" "H1_A2" ...
##  ..  ..$ : chr [1:6] "H1_A1" "H2_A1" "H3_A1" "H1_A2" ...
##  @ n_hypotheses: int 3
##  @ n_analyses  : int 2
##  @ column_names: chr [1:6] "H1_A1" "H2_A1" "H3_A1" "H1_A2" "H2_A2" "H3_A2"

For human readability, we can format the correlation matrix as a table.

corr_mat@matrix

##           H1_A1     H2_A1     H3_A1     H1_A2     H2_A2     H3_A2
## H1_A1 1.0000000 0.7627701 0.6666667 0.7071068 0.5393599 0.4714045
## H2_A1 0.7627701 1.0000000 0.6992059 0.5393599 0.7071068 0.4944132
## H3_A1 0.6666667 0.6992059 1.0000000 0.4714045 0.4944132 0.7071068
## H1_A2 0.7071068 0.5393599 0.4714045 1.0000000 0.7627701 0.6666667
## H2_A2 0.5393599 0.7071068 0.4944132 0.7627701 1.0000000 0.6992059
## H3_A2 0.4714045 0.4944132 0.7071068 0.6666667 0.6992059 1.0000000

References

Anderson, Keaven M, Zifang Guo, Jing Zhao, and Linda Z Sun. 2022. “A Unified Framework for Weighted Parametric Group Sequential Design.” Biometrical Journal 64 (7): 1219–39.

Chenguang Zhang, Yujie Zhao and Keaven Anderson