2.3.1 Fundamental method of sequential analysis

Sequential method has the capability to manage the overall Type I error in clinical trials and cumulative meta-analyses through the utilization of an alpha spending function, exemplified by the approach introduced by O’Brien & Fleming (1979). Based on required sample size (RSS) or required information size (RIS), sequential analysis applies the alpha spending function to allocate less significance level to early interim analyses, demanding more substantial evidence for declaring statistical significance. Critical boundaries for significance are determined by this function, becoming less stringent as more data accumulates. Sequential analysis uses these boundaries to monitor effects at each interim analysis, declaring statistical significance if the cumulative Z-score crosses them. This approach minimizes the risk of false positives in sequential analyses, where multiple interim analyses are conducted, while maintaining the ability to detect true effects.

Therefor, the essentials of sequential meta-analysis encompass various elements including study time, sample size, and cumulative Z-scores derived from cumulative meta-analysis following a temporal order. Additionally, integral to this analysis are the anticipated effects of intervention or exposure, which are either predetermined or established through prior estimates, along with their respective variances. Moreover, the formulation accounts for predefined probabilities associated with Type I error (denoted as \(\alpha\)) and Type II error (denoted as \(\beta\)). Leveraging this comprehensive set of parameters, particularly the anticipated effects and their associated variances in conjunction with \(\alpha\) and \(\beta\), this methodology facilitates the computation of the RIS and the establishment of alpha-spending monitoring boundaries. By charting the information size axis, the analysis offers a comprehensive visualization of the cumulative Z-scores and the accompanying alpha-spending monitoring boundaries (Wetterslev et al., 2008; Wetterslev et al., 2009; Wetterslev et al., 2017).

Within the realm of sequential method, the concept of the RIS holds paramount significance as it serves as the cornerstone for establishing spending monitoring boundaries, necessitated by adjustments to accommodate multiple interim analyses (Wetterslev et al., 2017). Essentially, sequential methods encompass considerations pertinent to sample size determination, albeit employing an approach divergent from conventional methodologies employed in hypothesis testing or estimation. The determination of the information size, delineating the RSS, hinges crucially upon various parameters including the anticipated effect size, statistical power, significance level, and any requisite adjustments tailored for multiple testing scenarios. Conventionally, the determination of the RSS encompasses an assessment of intervention or exposure effects, their variance, as well as the predefined probabilities associated with Type I error (\(\alpha\)) and Type II error (\(\beta\)). The formula governing the calculation of the RSS is as follows:

\[\begin{equation} RSS = 4 \times (Z_{1-\alpha/2} + Z_{1 - \beta})^2 \times \frac{\sigma^2} {\delta^2} \tag{2.1} \end{equation}\]

Where

\(\alpha\) represents the predefined overall probabilities of false positive.
\(\beta\) represents the predefined overall probabilities of false negative.
\(\delta\) refers to the assumed effect, representing either the minimal or expected meaningful effect.
\(\sigma^2\) refers to the variance that associated with the assumed effect.

In the context of dichotomous outcomes, the parameter \(\delta\) can be substituted with the effect size that a trial aims to address, denoted by \(\mu\), derived from the difference in proportions between the control and experimental groups, represented by \(P_{control} - P_{experiment}\). Here, \(P_{control}\) and \(P_{experiment}\) signify the proportions of observed events in the control and experimental groups, respectively. The variance \(\sigma^2\) is computed as \(P_{average} \times (1 - P_{average})\), where \(P_{average}\) is derived from \((P_{control} + P_{experiment}) / 2\) (Wetterslev et al., 2017). For continuous outcomes, the RSS can be determined as follows:

\[\begin{equation} RSS_{continuous} = 4 \times (Z_{1-\alpha/2} + Z_{1 - \beta})^2 \times \frac{MD^2} {SD^2} \tag{2.2} \end{equation}\]

Where

\(\alpha\) represents the predefined overall probabilities of false positive.
\(\beta\) represents the predefined overall probabilities of false negative.
\(MD\) refers to the assumed mean difference between two groups.
\(SD\) refers to the standard deviation that associated with the assumed mean difference.

Based on the basic formula for calculating RSS, RIS in sequential method could be calculated by using assumed pooled effect and the variance that associated with the assumed pooled effect (Wetterslev et al., 2017). Thus, the formula for the calculation of the RIS for meta-analysis in fixed-effect model is as follows:

\[\begin{equation} RIS_{fixed} = 4 \times (Z_{1-\alpha/2} + Z_{1 - \beta})^2 \times \frac{V_{fixed}} {\theta^2} \tag{2.3} \end{equation}\]

Where

\(\alpha\) represents the predefined overall probabilities of false positive.
\(\beta\) represents the predefined overall probabilities of false negative.
\(\theta\) refers to the assumed pooled effect, representing either the minimal or expected meaningful effect.
\(V_{fixed}\) refers to the variance that associated with the assumed pooled effect in fixed-effect model.

In meta-analysis employing the random-effects model, the calculation of the RIS follows a similar formula, but with a substitution of the variance associated with the assumed pooled effect by the variance specific to the random-effects model. RIS for meta-sequential analysis in random-effects model can be denoted as \(RIS_{random}\). For a conservative estimation of the RIS in meta-analyses featuring heterogeneity, it is advisable to incorporate a heterogeneity correction factor (Wetterslev et al., 2008), also referred to as the Adjusted Factor (AF) in Trial Sequential Analysis (TSA) (Wetterslev et al., 2009; Wetterslev et al., 2017). Obviously, the calculation of the heterogeneity correction factor is inherently tied to the heterogeneity statistics. The I-squared statistic (denoted as \(I^2\)) is particularly suited for this purpose, serving as a widely utilized measure for quantifying and assessing heterogeneity (Higgins & Thompson, 2002). \(I^2\)-based AF (denoted as \(AF_{I^2}\)) can be derived from the formula as follow:

\[\begin{equation} AF_{I^2} = \frac{1} {(1 - I^2)} \tag{2.4} \end{equation}\]

However, \(I^2\) may provide misleading estimates due to uncertainties in the sampling error could introduce bias or inaccuracies into the analysis. Alternative measures or adjustments that are less susceptible to issues related to sampling error estimation should be considered (Wetterslev et al., 2009). Diversity (denoted as \(D^2\)) has been proposed as a pivotal parameter in the computation of AF for the RIS in random-effects meta-analysis (Wetterslev et al., 2009; Wetterslev et al., 2017). It is derived from the variances observed in both the random-effects model (\(V_{random}\)) and the fixed-effect model (\(V_{fixed}\)). The formula for calculating \(D^2\) is as follows:

\[\begin{equation} D^2 = \frac{(V_{random} - V_{fixed})} {V_{random}} \tag{2.5} \end{equation}\]

Then, the calculation of \(D^2\)-based AF (denoted as \(AF_{D^2}\)) is similar to the formula of \(AF_{I^2}\) with replacement of \(I^2\) by \(D^2\). When addressing concerns regarding sampling error or bias, \(AF_{D^2}\) offers notable advantages over \(AF_{I^2}\) because \(D^2\) operates without the prerequisite of assuming a typical sampling error (usually denoted as \(\sigma^2\)). Besides, it accurately mirrors the relative variance expansion resultant from the between-trial variance estimate, and exhibits a propensity to diminish as the estimate decreases, even when applied to identical study sets (Wetterslev et al., 2009).

To provide a conservative estimation of the RIS for meta-analyses with heterogeneity, it is recommended to utilize an adjusted RIS. This can be achieved by multiplying \(RIS_{random}\) by the AF (Wetterslev et al., 2009). Subsequently, there are two the I-squared-adjusted information size (HIS) can be derived by further multiplying \(RIS_{random}\) by \(AF_{I^2}\). Similarly, the diversity-adjusted information size (DIS) is obtained by calculating \(RIS_{random}\) multiplied by \(AF_{D^2}\), as suggested by Wetterslev et al. (2009).

The following steps and syntax demonstrate how user can carry out sequential analysis. Figure 2.5 sequential analysis plot. This demonstration serves as a guide for users intending to conduct a prospective meta-analysis with predefined parameters such as Type I error (\(\alpha\)), Type II error (\(\beta\)), presumed effect size (relative risk reduction, denoted as \(RRR\), for dichotomous outcomes; mean difference, denoted as \(MD\), for continuous outcomes), an estimate of the variability associated with the presumed effect. However, this demonstration may not be applicable for meta-analyses conducted without such pre-specified parameters.

Step 1: Import data (example of the study by Fleiss 1993)

library(meta)
data("Fleiss1993bin")
dataFleiss1993bin <- Fleiss1993bin

Step 2: Do sequential analysis

DoSA(Fleiss1993bin, 
     source = study, 
     time = year,
     r1 = d.asp, 
     n1 = n.asp, 
     r2 = d.plac, 
     n2 = n.plac, 
     measure = "RR",
     PES = 0.1,
     RRR = 0.2,
     group = c("Aspirin", "Placebo"))

#> Summary of sequential analysis (main information)
#>  Acquired sample size: 28003
#>  Required sample size (heterogeneity adjusted): 21279
#>  Cumulative z score: -2.035
#>  Alpha-spending boundary: 1.709 and -1.709
#>  Adjusted confidence interval is not necessary to be performed.
#> 
#> Summary of sequential analysis (additional information)
#>  1. Assumed information
#>  1.1. Defined type I error: 0.05
#>  1.2. Defined type II error: 0.2
#>  1.3. Defined power: 0.8
#>  1.4. Presumed effect: 0.025
#>       (risks in group 1 and 2 were 9.90061431% (expected) and 12.37576788% respectively; RRR = 0.2)
#>  1.5. Presumed variance: 0.101
#> 
#>  2. Meta-analysis
#>  2.1. Setting of the meta-analysis
#>  Data were pooled using inverse variance approach in random-effects model with DL method.
#>  2.2. Result of the meta-analysis 
#>  Log RR: -0.113 (95% CI: -0.222 to -0.004)
#>  
#>  3. Adjustment factor 
#>  The required information size is calculated with adjustment factor based on diversity (D-squared). Relevant parameters are listed as follows.
#>  3.1. Heterogeneity (I-squared): 39.6%
#>  3.2. Diversity (D-squared): 76%
#>  3.3. Adjustement factor: 4.103

Step 3: Visualize sequential analysis

DoSA(Fleiss1993bin, 
     source = study, 
     time = year,
     r1 = d.asp, 
     n1 = n.asp, 
     r2 = d.plac, 
     n2 = n.plac, 
     measure = "RR",
     PES = 0.1,
     RRR = 0.2,
     group = c("Aspirin", "Placebo"),
     plot = TRUE)

Figure 2.5: An example for sequential analysis

2.3.2 Observed sequential analysis

Observed sequential analysis is recommended for those pooled analysis without pre-specified parameters for sequential analysis. In this situation, thus, Step 2 should use following syntax:

DoOSA(Fleiss1993bin, 
      source = study, 
      time = year,
      r1 = d.asp, 
      n1 = n.asp, 
      r2 = d.plac, 
      n2 = n.plac, 
      measure = "RR",
      group = c("Aspirin", "Placebo"))

#> Summary of observed sequential analysis (main information)
#>  Acquired sample size: 28003
#>  Optimal sample size (heterogeneity adjusted): 36177
#>  Cumulative z score: -2.035
#>  Alpha-spending boundary: 2.228 and -2.228
#>  Adjusted confidence interval is suggested to be performed.
#> 
#> Adjusted confidence interval based on type I error 0.0129503288310862: 
#> -0.252 to 0.025
#> 
#> Summary of observed sequential analysis (additional information)
#>  1. Observed information
#>  1.1. Defined type I error: 0.05
#>  1.2. Defined type II error: 0.2
#>  1.3. Defined power: 0.8
#>  1.4. Observed effect size 0.019
#>       (risks in group 1 and 2 were 10.47748283% and 12.37576788% respectively; logit transformation with Inverse method for pooling the proportion; RRR = 0.153)
#>  1.5. Observed variance: 0.101
#> 
#>  2. Meta-analysis
#>  2.1. Setting of the meta-analysis
#>  Data were pooled using inverse variance approach in random-effects model with DL method.
#>  2.2. Result of the meta-analysis 
#>  Log RR: -0.113 (95% CI: -0.222 to -0.004)
#>  
#>  3. Adjustment factor 
#>  The optimal information size is calculated with adjustment factor based on diversity (D-squared). Relevant parameters are listed as follows.
#>  3.1. Heterogeneity (I-squared): 39.6%
#>  3.2. Diversity (D-squared): 76%
#>  3.3. Adjustment factor: 4.103

2.3.2.1 Visualize basic plot of observed sequential analysis (example of the study by Fleiss 1993)

Observed sequential analysis could be visualized using the same function in terms of DoOSA() with argument TRUE for the parameter plot. The syntax and graph are shown as follows.

DoOSA(Fleiss1993bin,
      source = study,
      time = year,
      r1 = d.asp,
      n1 = n.asp,
      r2 = d.plac,
      n2 = n.plac,
      measure = "RR",
      group = c("Aspirin", "Placebo"),
      plot = TRUE)

Figure 2.6: An example for illustrating observed sequential analysis

2.3.2.2 Observed sequential analysis with colorful zones (example of the study by Fleiss 1993)

If users would like to show colorful zones of observed sequential analysis, two steps are recommended. Firstly, they need to store the output of function DoOSA() with argument TRUE for parameter Plot, then colorfur zones on the observed sequential analysis can be carried out using function PlotOSA() with argument TRUE for parameter lgcZone.

Step 1: Store observed sequential analysis with argument TRUE for parameter Plot

output <- DoOSA(Fleiss1993bin,
                source = study,
                time = year,
                r1 = d.asp,
                n1 = n.asp,
                r2 = d.plac,
                n2 = n.plac,
                measure = "RR",
                group = c("Aspirin", "Placebo"),
                SAP = TRUE)

Step 2: Show colorful zone on the observed sequential analysis with argument TRUE for parameter lgcZone

PlotOSA(output,
        lgcZone = TRUE)

Figure 2.7: An example for illustrating colorful zones on observed sequential analysis

2.3.2.3 Power curve of observed sequential analysis

If users would like to further illustrate power curve of the observed sequential analysis, two steps are recommended. Firstly, they need to store the output of function DoOSA() with argument TRUE for parameter SAP, then plot power curve of the observed sequential analysis using function PlotPower().

Step 1: Store output of observed sequential analysis with argument TRUE for parameter SAP

output <- DoOSA(Fleiss1993bin,
                source = study,
                time = year,
                r1 = d.asp,
                n1 = n.asp,
                r2 = d.plac,
                n2 = n.plac,
                measure = "RR",
                group = c("Aspirin", "Placebo"),
                SAP = TRUE)

Step 2: Power curve of observed sequential analysis

PlotPower(output)

Figure 2.8: An example for illustrating plot curve of observed sequential analysis