Therapeutic Index Function

Following the proposal of Filozof et al. (2017), Chow and Huang (2019b) studied the development of therapeutic index function for multiple endpoints in clinical trials, which is briefly described below.

Utility Function

In a clinical trial, suppose there are / study endpoints, denoted by

be the / clinical endpoints at baseline. The therapeutic index is defined as:

where со, = (con, coi2,..., ft),/) is a vector of weights with ft),, being the weight for ej with respect to index /,, and f (•) is a utility function, which is referred to as a therapeutic index function, to construct therapeutic index I, based on ft), and e. Generally,

binary, time-to-event endpoints), and ft)i( is pre-specified (or calculated by pre-specified criteria) and can be different for different therapeutic index Ij. Moreover, the therapeutic index function typically generates a vector of

index (Ji, 12,..., Ik)'; if К = 1, it reduces to a single (composite) index. As an example, consider

then f, is simply a linear combination of the endpoints; moreover, if

then Ij is the average over all the endpoints.

Selection of ω[sub(i)]

For the therapeutic index function given in (3.1), one of the important concerns is how to select the weights ft),. There might be various ways of specifying the weights (e.g., based on variabilities associated with the individual study endpoints), while we would like to propose selecting weights based on the p-values observed from the individual study endpoints. In clinical trials, the p-values are indicators of the levels of substantial evidences regarding the safety and effectiveness of the test treatment under investigation provided by the individual endpoints. Specifically, denote

as the treatment effect assessed by the endpoint ej. Without the loss of generality, 6j is tested by the following hypotheses:

where <5,, j = 1,..., / are the pre-specified margins. Under some appropriate assumptions, we can calculate the p-value p, for each H0j based on the sample of ej and the weights ft), can be constructed based on

That is,

which is reasonable since each p value indicates the significance of the treatment effect based on its responding endpoint; thus, it is possible to use all the information available to construct effective therapeutic indexes. Note that (Ojj (•) should be constructed such that high value of сщ is associated with low value of pr For example,

Determination of f[sub(i)](.)

Another important issue is how to select the utility function /,{■) for the construction of the therapeutic index. In practice, f (•) could be linear or nonlinear, or with more complicated forms. For simplicity, we consider /;(•) as a linear function here. Thus, (4.1) reduces to

Assumption for the Distribution of e

In order to study the statistical properties of the developed therapeutic index described in (4.1), we need to specify the distribution of e. For simplicity, we

assume e follows the multidimensional normal distribution N(0, X), where

and

with

Statistical Evaluation of the Therapeutic Index

Evaluation Criteria

Although l, given that y given that /, is informative. Particularly, we are interested in the following two conditional probabilities:

and

Intuitively, we would expect that рц to be relatively large given that et is informative since /, is a function of e;, especially when relatively high weight is assigned to ер, on the other hand, рц could be small even if 1, is predictive since the information contained in 1, may be attributed to another endpoint rather than other er

In what follows, equations (4.5) and (4.6) will be derived under the assumption that (i) the specified weights со, based on p-values, and (ii) the distribution of e and the functions/(•) are described in the previous section.

Derivation of Pr (I[sub(i)]| e[sub(j)]) and Pr (e[sub(j)| I[sub(i0])

Suppose n subjects are independently and randomly selected from the population for the clinical trial. For each baseline endpoint e, and hypothesis H0y, a test statistic is constructed based on the observations of the n subjects and the corresponding p-value p, is calculated. e} is informative and is equivalent to Cj > Cj for some pre-specified critical value cy/ based on <5„ significance level a, and the variance of er The estimate of the therapeutic index I, in (4.4) can be accordingly constructed as

where

and

is calculated based on the p-values on and

Ii is informative if I, > d, for some pre-specified threshold d,. Thus, (4.5) and (4.6) become

and

Without the loss of generality, suppose e is the vector of sample means, then e follows the multidimensional normal distribution N (0, X/n) based on the normality assumption of e. Moreover, I, follows the normal distribution N(

and

Further, [ej, Ij) jointly follows a binormal distribution N (p, Г/и), where and

where 1, is a J dimensional vector of 0 except the jth item, which equals 1, and thus

Thus, the conditional probabilities (4.8) and (4.9) become and

Moreover

where Ф(х) and vP(.r, y, p) denote the cumulative distribution functions for standard single variate normal and bivariate normal distributions, respectively. Note that both conditional probabilities (4.10) and (4.11) depend on the parameters 0, X; the sample size n; the number of baseline endpoints /; the pre-specified weights to,; and the pre-specified thresholds cy, d,, which further depend on the hypothesis testing margins <5y and pre-specified type I error rate(s), among others. Intuitively, there are not simple formulas for (4.10) and (4.11) that can be derived directly. Although methods like Taylor expansion may be employed to approximate (4.10) and (4.11), it is still nontrivial and could be quite complicated. However, note that Ф(х) is monotonic increasing, based on (4.12) we have

Moreover, we assume

conventionally, and d, is a linear combination of c,s, i.e.,

Then (4.13) is further expressed as

where

equals ft), except the /'th item equals 0. To obtain more insights of (4.15), we assume J = 2, К = 1, and focus on j = 1 without the loss of generality. Then, the last inequality in (4.15) can be simplified as

where ft)i and co2 are the weights for the two endpoints, respectively, with (0i + (02 = 1, and

and p is the correlation coefficient of the two endpoints. Obviously, (4.16)

depends on the variabilities of the endpoints and their correlation, the

underlined effect sizes of both endpoints, the weights, and the sample size.

We illustrate several special scenarios of (4.16) in Table 4.4.

From Table 4.3, we can see a remarkable situation that when p = l, a, = -o2,

т

whether pUj is greater than p2ij depends on whether the underlined effect size AOi is less than — Ав2 only, regardless of the weights. For other situations, the relation between рц and p2ij varies for different combinations of weights, variabilities and correlations, underlined effect sizes, and sample sizes.

Remarks

In this section, for simplicity, we assume different single endpoints are of the same data type, i.e., continuous endpoint. In practice, however, different endpoints of different data types may be considered in a clinical trial like a cancer clinical trial. In this case, the development of therapeutic index is much more difficult, if it is not impossible, because it is not easy to figure out the joint distribution of endpoints with different data types (e.g., a continuous endpoint, a binary endpoint, and/or a time-to-event endpoint) without some plausible assumptions.

For illustration of the idea of the use of the developed therapeutic index in clinical trials, we consider data transformations such that all the endpoints are continuous in order to overcome these difficulties. In this case, the multinormal approximation can be assumed for the joint distribution of the endpoints for the evaluation of the developed therapeutic index. Specifically, for continuous endpoint, we considered standardization of the data, i.e., subtract the mean and then divide it by the standard deviation.

TABLE 4.4

Illustrated Inequalities (4.16) with Respect to Different Parameter Settings

Parameters

Inequality (4.16)

For binary endpoint, treat the binary endpoint as the response variable and fit a regression model (e.g., logistic model, probit model, or log-log model, among others) with some reasonable independent variable(s) (with input from clinician). Then, standardize the estimated success probabilities. For time-to-event endpoint, we may either subtract the median and then divide it by the standard deviation of the median, or subtract the mean and then divide it by the standard deviation of the mean. The transformed endpoints are then used as the surrogate endpoints to construct the therapeutic indices.

In what follows, the above idea is illustrated using a numerical example based on the real practices.

A Numerical Example

In this section, we will demonstrate how to develop a therapeutic index that combines a binary endpoint and a survival endpoint based on the simulated patient data, which mimic a recent cancer trial.

Simulate Patient Data

For illustration purpose, we simulated patient data which contain a binary endpoint and a survival endpoint using parameters with respect to patients with PD-Ll-positive tumors reported in a recent cancer trial (Motzer et al., 2019).

Without the loss of generality and for simplicity, we consider a single-arm study with n = 300 patients. Assume that the underlying success probability for the selected binary endpoint is about pb = 0.55 and the median PFS for the selected time-to-event endpoint is about ts = 13.8 month. Denote the binary endpoint as ex and the survival endpoint as e2. Thus, e = (eu e2)'. The procedure for the generation of the patient data is summarized below.

Step 1. Generate a two-dimensional random vector X = (Xb X2) from a binormal distribution

where

and

Step 2. Generate a random error vector £ = {£, £2)' from a binormal distribution

Step 3. Calculate

and generate the binary endpoint e: from the Bernoulli distribution with the success probability of e.

Step 4. Calculate e2 = exp(X2 + £2) and generate a censoring variable ec with

Let

and

The survival endpoint is e2 = {T, C}.

For the generated patient data, the average of ег is about 0.53, which is close to рь; the median of e2 is equal to ts with a censor rate about 29.3%. Moreover, the correlation between the underlined success probability e of the binary endpoint and the survival endpoint e2 is about 0.14, which can be adjusted by modifying o2 in the above procedure. Assume the random variable Xj is also obtained such that it can be used as the explanatory variable to fit the regression model for e,. Thus, we generated a data set which is comprised of a binary endpoint e1f a survival endpoint e2, and a continuous variable X, to use as an explanatory variable for fitting a logistic regression model.

Data Transformation, Normal Approximation, and Illustration

Fitting the logistic regression model with the explanatory variable Xb we have

where pb is the underlying success probability of ex. The fitted value of success probability has a sample mean of 0.53 and a sample standard deviation of 0.50, and its standardized form is denoted as ex.

For survival data, median is more commonly used than mean. Flowever, the variance of the median is intractable (Rider, 1960) such that median may not be a good option for data standardization. Thus, the mean is adopted for data standardization of the survival endpoint e2. The same mean and standard deviation of e2 are 15.39 and 5.43, respectively, and the standardized form of e2 is denoted as 2.

The two standardized endpoints c, and e2 are then assumed to approximately follow the binormal distribution with mean zero and variance 1, and their correlation is estimated from the standardized observations (about 0.17 in the simulated patient data).

Note that the two conditional probabilities (4.8) and (4.9), and рц > рц is equivalent to that probability (4.16), hold when there are two endpoints with joint normal assumption (or approximation). To illustrate an instance of (4.16), assume <5i = 0.39 and 82 = 13.8. Note that в = 0.53 and в2 = 15.39; thus, Авх = #i - <5, = 0.14 and Ав2 = 92-82 = 1.59. Moreover, ox = 0.50, o2 = 5.43, и = 300, p = 0.17, and z« = 1.96 given a = 0.025. Note that (4.16) can be expressed as

where e = ^ Ав2 ~^j=o2 j|/^ A0X ~^j=ax j, given that Авх ~^j=0 > 0. On the

one hand, if we assume equal weights, i.e., (0 =(02 = 1/2, the left-hand side of (4.18) is 10.78 and (4.18) holds. On the other hand, recall the hypothesis testing in (4.4), the ratio of p-values of the two alternative hypotheses Ha 1: 6, > 8{ and

1 1 1

H„2 62 > S2 is px/p2 = 3.38/2.03; if we let (0■. =/S' —, then cox = 0.37 and

Pi TtPi

co2 = 0.63, the left-hand side of (4.18) is 10.96 and (4.18) holds again. Thus, the hypothetical example supports the intuition of pXij > p2ij, i.e., the conditional probability that the therapeutic index l, is informative given that one original endpoint ej is informative is larger than the probability that one original endpoint ej is informative given that the therapeutic index 1, is informative.

Concluding Remarks

In clinical trials, endpoint selection is always a concern, especially when multiple endpoints are available. Based on the available endpoints, a number of derived endpoints (including individual endpoints, co-primary endpoints, and composite endpoints) may be obtained. In practice, it is often a concern which (derived), which can best inform the disease status and/or treatment effect, should be used for the evaluation of the safety and effectiveness of the test treatment under investigation as some endpoints may be achieved and some may not be. Alternatively, the concept of therapeutic index that is able to combine and fully utilize the information collected from the individual and derived endpoints is developed. In this chapter, for simplicity and illustration purpose, a therapeutic index was developed under the assumption that (i) different endpoints are of the same data type, (ii) the utility function is linear, and (iii) the selection of weights is based on the observed p-values of individual endpoints. The results indicate that the performance of the developed therapeutic index is better than those of the individual endpoints and/ or co-primary endpoints in terms of false-positive and false-negative rates.

For the case where different endpoints are of different data types (e.g., continuous, binary, or time-to-event endpoint), as discussed in Section 2, to calculate the conditional distributions of (4.5) and (4.6), it is necessary to obtain the joint distribution of the baseline endpoints and the therapeutic indices. This is, however, difficult if not impossible to obtain. To overcome these difficulties, it is suggested appropriate data transformation be considered for obtaining normal approximation in order to use the theoretical results derived in this chapter. The idea of data transformation for an application in cancer trial is illustrated in Section 4.4.

It should be noted that for studying the joint distribution of different endpoints, there are results available in the literature. For example, for studying the joint distribution of two binary endpoints eltU eb2 with success probabilities

and

their correlation coefficient is (Leisch et al., 1998)

When there are more than two binary endpoints, their joint distribution and pairwise correlations can be similarly extended. Moreover, the marginal and conditional distributions of a multivariate Bernoulli distribution are still Bernoulli (Dai et al., 2013). However, the distributions of therapeutic indices constructed based on the binary endpoints can be very complicated. One simple exception is that the index follows a binomial distribution given that it is the sum of mutually independent binary endpoints with the same success probability. On the other hand, for time-to-event endpoints, the joint distribution and correlation structure can be more complicated than those of binary endpoints. Although there are a few methods to estimate the correlation of bivariate failure times under censoring, such as the normal copula approach (NCE) and the iterative multiple imputation (IMI) method (Schemper et al., 2013), it is still very difficult to derive the joint distribution of survival endpoints, which often require strong model assumptions, in addition to the complexity of the joint and conditional distributions of the survival endpoints and the indices.

 
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