Monitored Analyses for a Continuous Outcome
Source:vignettes/analyses_continuous.Rmd
analyses_continuous.RmdThis vignette demonstrates how to use impart to conduct
interim and final analyses in group sequential or information monitored
designs. This builds upon the terminology and concepts in the vignettes
on study design and information monitoring. To see all available
vignettes in impart, use the vignettes command:
vignette(package = "impart")| Title | Item |
|---|
Setting Up a Monitored Design:
The design parameters in this vignette will be the same as in the previous vignettes:
# Universal Study Design Parameters
minimum_difference <- 5 # Effect Size: Difference in Means of 5 or greater
alpha <- 0.05 # Type I Error Rate
power <- 0.9 # Statistical Power
test_sides <- 2 # Direction of Alternatives
# Determine information required to achieve desired power at fixed error rate
information_single_stage <-
impart::required_information_single_stage(
delta = minimum_difference,
alpha = alpha,
power = power
)
# Group Sequential Design Parameters
information_rates <-
c(0.50, 0.75, 1.00) # Analyses at 50%, 75%, and 100% of the Total Information
type_of_design <- "asOF" # O'Brien-Fleming Alpha Spending
type_beta_spending <- "bsOF" # O'Brien-Fleming Beta Spending
# Set up group sequential testing procedure
trial_design <-
rpact::getDesignGroupSequential(
alpha = alpha,
beta = 1 - power,
sided = 2,
informationRates = information_rates,
typeOfDesign = type_of_design,
typeBetaSpending = type_beta_spending,
bindingFutility = FALSE
)
# Inflate information level to account for multiple testing
information_adaptive <-
impart::required_information_sequential(
information_single_stage = information_single_stage,
trial_design = trial_design
)
# Initialize the monitored design
monitored_design <-
initialize_monitored_design(
trial_design = trial_design,
null_value = 0,
maximum_sample_size = 280,
information_target = information_adaptive,
orthogonalize = TRUE,
rng_seed_analysis = 54321
)The data used in this example will also be the same:
impart::example_1. Specifically, the data has been reverted
to particular points in the study at which the information thresholds
have been met:
-
example_1_ia_1: Data for Interim Analysis 1 -
example_1_ia_2: Data for Interim Analysis 2 -
example_1_final: Data for Final Analysis
# Data for Interim Analysis 1
example_1_ia_1 <- impart::example_1_ia_1
head(example_1_ia_1)
#> .id x_1 x_2 x_3 x_4 tx .e .r_1 .t_1
#> 1 1 2.0742970 0.1971432 -0.8425884 0.2794844 0 2.24846 1 25.01538
#> 2 2 0.2165473 -0.7384296 0.1315016 -1.2419134 1 11.05565 0 55.05565
#> 3 3 0.8294726 0.4997821 1.6932555 -0.4063889 0 16.96591 0 60.96591
#> 4 4 -1.0206893 -0.2189937 -1.7719120 0.1936013 1 25.13396 1 59.84544
#> 5 5 -0.0417332 0.9282685 0.8078133 0.9317145 0 50.07301 1 75.94952
#> 6 6 0.7275778 1.1756811 0.0226265 -0.2556343 1 50.93935 1 80.29181
#> y_1 .r_2 .t_2 y_2 .r_3 .t_3 y_3 .r_4 .t_4
#> 1 1.591873 1 56.62636 -4.535711 1 98.51499 13.98543 1 133.0050
#> 2 NA 0 85.05565 NA 0 115.05565 NA 0 145.0556
#> 3 NA 0 90.96591 NA 0 120.96591 NA 0 150.9659
#> 4 1.212620 1 81.58577 -4.533776 1 127.78659 11.17615 1 154.3419
#> 5 8.655326 1 111.21967 6.970372 1 143.00338 17.62329 1 181.4162
#> 6 6.902055 1 114.96781 17.381316 1 151.34249 -2.42570 1 177.3846
#> y_4
#> 1 -1.320242
#> 2 NA
#> 3 NA
#> 4 -6.629545
#> 5 9.126240
#> 6 3.549977Interim Analysis 1
Before conducting the first interim analysis, the information level should be above the pre-specified threshold: the smoothed trajectory can be used to mitigate random variation in the information level.
Analysts will need to specify estimation_function, the
function used to compute the estimate, and
estimation_arguments, a list of arguments aside from the
data needed for this computation. Estimators may have a variance
correction factor that can be computed from the analysis parameters:
this can be specified using the correction_function
argument.
For the standardization estimator computed by
standardization(), the corresponding small-sample variance
correction is standardization_correction(): this adjusts
the variance according to the sample size in each arm and the number of
parameters in each regression model.
# Obtain time of last event
last_event <-
example_1_ia_1[, c(".e", ".t_1", ".t_2", ".t_3", ".t_4")] |>
unlist() |>
max(na.rm = TRUE) |>
ceiling()
example_1_ia_1_prepared <-
prepare_monitored_study_data(
data = example_1_ia_1,
study_time = last_event,
id_variable = ".id",
covariates_variables = c("x_1", "x_2", "x_3", "x_4"),
enrollment_time_variable = ".e",
treatment_variable = "tx",
outcome_variables = c("y_1", "y_2", "y_3", "y_4"),
outcome_time_variables = c(".t_1", ".t_2", ".t_3", ".t_4"),
# Observe missingness 1 week after target study visit
observe_missing_times = c(30, 60, 90, 120) + 7
)
data_ia_1_trajectory <-
information_trajectory(
prepared_data = example_1_ia_1_prepared,
monitored_design = monitored_design,
estimation_function = standardization,
estimation_arguments =
list(
estimand = "difference",
y0_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
y1_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
family = gaussian,
treatment_column = "tx",
outcome_indicator_column = ".r_4"
),
correction_function = standardization_correction,
orthogonalize = TRUE,
n_min = 50,
n_increment = 3,
rng_seed = 23456,
control = monitored_analysis_control()
)
data_ia_1_trajectory
#> times randomization y_1 y_2 y_3 y_4 information information_lag_1
#> 1 551.8682 90 69 57 53 50 0.1380683 NA
#> 2 555.9573 92 70 60 54 50 0.1339839 0.1380683
#> 3 595.6751 96 77 63 58 53 0.1425721 0.1339839
#> 4 630.7461 100 79 70 64 56 0.1482429 0.1425721
#> 5 634.3194 100 79 70 64 59 0.1711911 0.1482429
#> 6 658.6659 109 82 72 69 62 0.1696971 0.1711911
#> 7 662.6718 110 82 72 69 65 0.1665782 0.1696971
#> 8 673.9776 113 87 74 70 65 0.1633559 0.1665782
#> 9 701.1256 117 96 80 71 68 0.1805331 0.1633559
#> 10 717.5104 119 96 82 73 71 0.2098995 0.1805331
#> 11 753.7782 126 101 91 83 74 0.2179450 0.2098995
#> 12 758.3298 128 101 91 85 77 0.2021433 0.2179450
#> 13 778.4102 131 105 94 88 80 0.2070840 0.2021433
#> 14 781.6646 131 106 94 89 83 0.2181654 0.2070840
#> information_change information_pct_change information_fraction
#> 1 NA NA 0.3028952
#> 2 -0.004084420 -3.0484414 0.2939348
#> 3 0.008588275 6.0238099 0.3127758
#> 4 0.005670759 3.8253158 0.3252163
#> 5 0.022948176 13.4050071 0.3755602
#> 6 -0.001493985 -0.8803831 0.3722827
#> 7 -0.003118900 -1.8723338 0.3654404
#> 8 -0.003222324 -1.9725794 0.3583713
#> 9 0.017177195 9.5147089 0.3960547
#> 10 0.029366413 13.9907033 0.4604790
#> 11 0.008045561 3.6915549 0.4781294
#> 12 -0.015801736 -7.8170963 0.4434634
#> 13 0.004940738 2.3858613 0.4543024
#> 14 0.011081351 5.0793349 0.4786128Once the trajectory has been computed, it can be smoothed and plotted:
plot(
information ~ y_4,
data = data_ia_1_trajectory
)
abline(
lm(
formula = information ~ y_4,
data = data_ia_1_trajectory
),
lty = 1
)
# Requires `deming` package
abline(
deming::theilsen(
formula = information ~ y_4,
data = data_ia_1_trajectory
),
lty = 3
)
abline(
h = monitored_design$original_design$information_thresholds,
lty = 2
)
Once the information level has been confirmed to be above the threshold, conducting the analysis is similar to computing information:
interim_analysis_1 <-
monitored_analysis(
data = example_1_ia_1,
monitored_design = monitored_design,
estimation_function = standardization,
estimation_arguments =
list(
estimand = "difference",
y0_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
y1_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
family = gaussian,
treatment_column = "tx",
outcome_indicator_column = ".r_4"
),
correction_function = standardization_correction
)
interim_analysis_1$interim_analysis_1$decision
#> [1] "Continue"
interim_analysis_1$interim_analysis_1$decision_data
#> test_statistic efficacy futility
#> 1 2.044012 2.930993 0.4236436
#> 2 NA 2.361025 1.2803287
#> 3 NA 2.014278 NA
interim_analysis_1$interim_analysis_1$information_fraction_orthogonal
#> estimates
#> estimates 0.5095383Interim Analysis 2
All subsequent analyses are identical in syntax: a new dataset is
provided, and the result of the previous analysis is passed using the
monitored_design argument.
example_1_ia_2 <- impart::example_1_ia_2
interim_analysis_2 <-
monitored_analysis(
data = example_1_ia_2,
monitored_design = interim_analysis_1,
estimation_function = standardization,
estimation_arguments =
list(
estimand = "difference",
y0_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
y1_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
family = gaussian,
treatment_column = "tx",
outcome_indicator_column = ".r_4"
),
correction_function = standardization_correction
)
interim_analysis_2$interim_analysis_2$decision
#> [1] "Continue"
interim_analysis_2$interim_analysis_2$decision_data
#> test_statistic efficacy futility
#> 1 2.044012 2.930993 0.4311741
#> 2 2.190967 2.281878 1.4392127
#> 3 NA 2.025798 NA
interim_analysis_2$interim_analysis_2$information_fraction_orthogonal
#> [1] 0.5095383 0.7921346Final Analysis
The syntax is identical for the final analysis:
example_1_final <- impart::example_1_final
final_analysis <-
monitored_analysis(
data = example_1_final,
monitored_design = interim_analysis_2,
estimation_function = standardization,
estimation_arguments =
list(
estimand = "difference",
y0_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
y1_formula = y_4 ~ x_1 + x_2 + x_3 + x_4,
family = gaussian,
treatment_column = "tx",
outcome_indicator_column = ".r_4"
),
correction_function = standardization_correction
)
final_analysis$final_analysis$decision
#> [1] "Efficacy: Upper"
final_analysis$final_analysis$decision_data
#> test_statistic efficacy futility
#> 1 2.044012 2.930993 0.271853
#> 2 2.190967 2.281878 1.199716
#> 3 2.199126 2.048468 NA
final_analysis$final_analysis$information_fraction_orthogonal
#> [1] 0.5095383 0.7921346 1.0956921