Application of Dynamic Linear Models to Random Allocation Clinical Trials with Covariates
AApplication of Dynamic Linear Models to Random AllocationClinical Trials with Covariates
Albert H. Lee III
Virginia Commonwealth University, Richmond, Virginia, USA
ARTICLE HISTORY
Compiled September 24, 2020
ABSTRACT
A recent method using Dynamic Linear Models to improve preferred treatmentallocation budget in random allocation models was proposed by [1]. However thismodel failed to include the impact covariates such as smoking, gender, etc, had onmodel performance. The current paper addresses random allocation to treatmentsusing the DLM in Bayesian Adaptive Allocation Models with a single covariate. Weshow a reduced treatment allocation budget along with a reduced time to locatepreferred treatment. Furthermore, a sensitivity analysis is performed on mean andvariance parameters and a power analysis is conducted using Bayes Factor. Thispower analysis is used to determine the proportion of unallocated patient budgetsabove a specified cutoff value. Additionally a sensitivity analysis is conducted oncovariate coefficients.
KEYWORDS
Bayes Factor, Dynamic Linear Model, Random Allocation, Clinical Trials, TimeSeries
1. Introduction
Clinical trials are popular research methods used to determine a preferential treat-ment when more than one possible treatment exists by reducing between group bias.These treatments are randomly assigned to groups of patients receiving a particulartreatment. According to Zelen [2] these groups “are as similar as possible except forthe administered treatment whereby the groups are decided through randomization”.Randomization procedures in clinical trials have been extensively researched, and whileassigning an equal number of patients to each treatment is the most common method,ethical issues using this method were discussed by [3].Ideally, a sequential allocation of patients to treatments through a random methodwhich skews patients to the most effective treatment while retaining a fully randomizedprocess is preferred. This process, known as random allocation, has been extensivelyresearched. This research includes the early works of [4–6]. Further research led tothe Play the Winner Rule of [7], and its modifications made by [8]. Additional worksinclude those of [3,9]. A Bayesian approach was used by [10] to compare the works ofboth [11] and [12] for binary outcomes.
CONTACT Albert H. Lee III. Email: [email protected] a r X i v : . [ s t a t . A P ] S e p nother method which has been used in random allocation processes involves adap-tively allocating subjects between treatments through the Bayesian Adaptive Design.Here, Bayesian updating methods are used to allocate subjects to treatments. Thisdesign involves transforming updated information into prior information through re-peated updating. According to Thall and Wathen [12] this ability provides “a nat-ural framework for making decisions based on accumulating data during a clinicaltrial”. Likewise, Berry [13] indicated Bayesian updating ability provided “the abilityto quantify what is going to happen in a trial from any point on (including from thebeginning), given the currently available data”. There has been much research donein this area including the works of [14–17]. Additionally, the works of Sabo [18] illus-trated a Bayesian approach to create what he termed “Decreasingly Informative Prior”information. This was used to evaluate the adaptive allocation performance when us-ing binary variables. Recently, [1] used a Dynamic Linear Model approach to randomallocation, and demonstrated reduced time and patient budget used to identify thepreferred treatment.Often with human subjects however, there exist covariates such as smoking, age,or sex to mention a few, which may impact the response. It is therefore imperativeto include these covariates, provided they exist, when randomly allocating subjects totreatments. The literature for covariate influenced adaptive allocation is quite sparse.The idea of D A optimality was discussed by [19] for a biased coin design method, how-ever, this did not include the random allocation. The works of [20] compared severalrandom allocation methods, however, they did not include any covariate influences.Although [21] used normal responses, they failed to consider the influence of covariates.A covariate adjusted method was proposed by [22] for the Doubly Adaptive BiasedCoin Design, however, it looked at the variability reduction rather than the alloca-tion methods. An examination of the asymptotic properties along with a theoreticalexamination may be seen in [23], however, as with the previous authors, no randomallocation was completed. However, [24] were able to use their method when covariateswere present.When investigating the impact of a single covariate, let y patients enter a randomallocation study sequentially at different times each with a single covariate x . These y t patients and their x t covariates may then be considered components of a time se-ries. Additionally, patient budget size is set to be a total of N patients during thetrial such that T is the index set for patient y t with covariate x t measured in a totalof N patients. As these sequentially entering y t patients enter the allocation studyupdating procedures provide additional allocation information regarding treatmenteffectiveness toward the better treatment. Using a Bayesian Adaptive Design createsa Bayesian Learning Method, whereby information regarding the better treatmentis learned as more patients enter the study. This information may then be appliedto entering patients. For instance, increased information regarding the better treat-ment may be applied to patient y through the updated information which occurredthrough patient y . Thus more information is known at patient y than at patient y , and as information is updated, the Bayesian design learns the better treatment.The aforementioned allocation method is capable of allocating subjects to treatmentswhen these covariates exist. 2 ayesian Methods Numerous works exist whereby Bayesian methodologies have been applied. Some ofthese works include theoretical texts by [25] who applies Bayesian ideas to samplingmethodologies. Additional works include those of [26] who illustrates how to applyBayesian methods using the R programming language in combination with a theoreti-cal overview. A discussion on Bayesian Loss functions may be found in [27], while [28]chapter 1 provides an additional introduction.The basic premise surrounding Bayesian methods is known as Bayes rule, namedafter Rev. Thomas Bayes. The idea posed by Bayes was p ( θ | y ) = p ( θ ) p ( y | θ ) /p ( y ) (1)where p ( θ | y ) represents the posterior distribution of θ given the known y data. Likewise p ( θ ) p ( y | θ ) ∝ p ( θ, y ) (2)Here, p ( θ ) is defined to be the prior probability of the parameter θ by Gelmanet al. [25] and p ( θ, y ) is the conditional probability involving θ and y . Furthermore,by conditioning on the known y data, the sampling distribution probability, p ( y | θ )provides the posterior probability (See [25] for more details.) Additional work usingthese ideas in the application of time series data has been done by [28]. Yet, once theposterior probability p ( θ | y ) has been calculated, it may then be used as a new priorprobability and the process repeated, with the Bayesian Updating learning along theway.The Dynamic Linear Model (DLM) of Harrison and West [29] uses this updatingprocess to create a Bayesian Learning Process. The learning ability created by this up-dating provides a useful mechanism whereby the DLM may forecast the y observationssuch that Y t = F (cid:48) t θ t + ν t (3) θ t = G t θ t − t + ω t where ν t ∼ N (0 , V t ) (4) ω t ∼ N (0 , W t ) As defined both by Harrison and West [29], and previously in [1] θ t represent theforecast parameter F t where F t is a known n × r matrix of independent variables, G t is a known n × n system matrix, W t is a known n × n evolution variance matrix, and V t is a known r × r observational variance matrix.3he prior forecast parameter θ t is found by noting ( θ t − | D t − ) ∼ N ( m t − , C t − ) for some mean m t − and variance matrix C t − . The prior for θ t may be seento be ( θ t | D t − ) ∼ N ( a t , R t ) whereby a t = G t m t − with R t = G t C t − G (cid:48) t + W t .The one step ahead forecast is calculated as ( Y t | D t − ) ∼ N ( f t , Q t ). Here, f t isthe current treatment allocation for patient y , while Q t is the forecast allocationvariance for patient y . The posterior for θ t relies on ( θ t − | D t − ) ∼ N ( m t , C t ) Furthermore, m t = m t − + A t e t , where m t represents the current mean matrix, C t = R t − A t Q t A (cid:48) t where C t is the current variance matrix, A t = R t F t Q − t where A t is the adaptive coefficient, and e t = Y t − f t represents the error term. Random Allocation Methods
There have been several methods used to minimize allocation responses. One suchsolution was proposed by [20], who suggested using w A = Q At √ f Bt Q At √ f Bt + Q Bt √ f At if ( f A t < f B t | Q At √ f Bt Q Bt √ f At > Q At √ f Bt Q At √ f Bt + Q Bt √ f At if ( f A t > f B t | Q At √ f Bt Q Bt √ f At < Otherwise (5) w B = 1 − w A to determine the optimally weighted allocation value solution. This solution wasshown by [21] to be problematic because it was possible for f A t or f B t to be negative,therefore Biswas and Bhattacharya [21] proposed their optimal solution ω A = Q A t (cid:112) f B t Q A t (cid:112) f B t + Q B t (cid:112) f A t (6) ω B = 1 − ω A where γ A = Φ f A t − f B t (cid:113) Q A t + Q B t , γ B = Φ f B t − f A t (cid:113) Q A t + Q B t Recently, [30] examined how a Decreasingly Informative Prior distribution impactedthe allocation using each of these equations. The DLM was applied by [1] and used tocompare the allocation results between the two equations using no covariate. In thecurrent work a covariate is included and a comparison made. Because the DLM is anupdating method at each value, the values for each of f A t , f B t , Q A t , Q B t will changeat each iteration, leading to different weight values based on the starting values. Forthis application, the covariate was generated as a Uniform (0,1) random variable. Alogrithm
To generate the allocation values(1) Initiate the DLM for µ A , µ B , ω t , C t A , C t B , Q t A , Q t B .42) Identify x t and calculate predicted values and variances f A t ( F t = [1 , f B t ( F t = [1 , , x t ]), Q A t and Q B t (3) Compute w A and w B (4) Sample a Uniform(0,1) random variable U and compare w A (5) If w A < U , allocate to Treatment A ( F t = [1 , F t = [1 , , x t ])(6) Conduct experiment and observe y t (7) Update the DLM and return to step 2 Simulation Study
The seven scenarios in Table 1 were investigated by [30] using the Decreasingly Infor-mative Prior and then by [1] using the DLM and including a covariate. Each grouprandomly allocated each scenario through 1000 simulations, and the treatment allo-cation probabilities, total number of allocations in each treatment group, and totalnumber of successes was recorded. However, the current authors have only includedthe treatment allocation associated with the preferred treatment and these may beseen in Table 2. The Decreasingly Informative Prior Method of [30] utilized manualiterations for each iteration. This lead to an sizable number of simulated calculationruns which lead to considerable completion times. The method of [1] was applied witha covariate added to the model and these times were greatly reduced. Each scenariowas run using R Studio version 1.2.1335 on an ACER computer with an AMD Ryzen 52500U with Radeon Vega Mobile Gfx 2.00 GHz processor and 8.00 GB of RAM usingWindows 10. The mean run time was approximately 120.259 seconds to completion.The lowest run time to completion was 60.960 seconds using the budget size N = 34.The highest run time to completion was 120.690 seconds using budget size N = 200, Table 1.
Simulation Scenarios
Scenario Differences Standard Deviation Planned Sample Budget1 0 20 1282 10 15 743 10 20 1284 10 25 2005 20 20 346 20 25 527 20 30 74An analysis was conducted using each of the values in Table 1 and the allocationvalues may be observed in Table 2. The mean number of allocations was obtainedusing each method. Notice the mean allocation using equation 5 attributed to [20] was63.542, which is as expected, given the probability of allocation to Treatment A was 0.5.The equal treatment allocation proportion for µ B = 0, standard deviations = 20 andbudget size N = 128 may be observed in Figure 1a. However, when the unequal methodof [21] in equation 6 was applied to the same parameters, the mean number appliedto Treatment A is 96.716, while the mean number allocated to Treatment B is 31.284.The proportion results for equation 10 may be observed in Figure 1b. Here the meanallocation proportion for treatment A was 0.654, while mean allocation proportion totreatment B was 0.346. Additionally it is important to note the immediate convergenceto either 0 or 1 using these values. Under the methods of [20,21,30], the smaller value5 lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s (a) Equal Allocation. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s (b) Unequal Allocation. Figure 1.
Comparison Between Equal and Unequal Allocation With Covariates. was taken to be the better allocation, therefore, it appears as though Treatment B isthe favorable treatment.
Table 2.
Treatment Group Mean Sample Size. Italicized values indicate Treatment B was selected
Mean SD Sample Equation 5 Equation 6Difference Budget Allocation Allocation B0 20 128 63 .
10 15 74 36 .
10 20 128 63 .
10 25 200 99 .
20 20 34 16 .
20 25 52 25 .
20 30 74 36 . Similar to [1] the mean, system variance and observational variance were varied todetermine treatment allocation weight behavior. The modifications made to these pa-rameter values will aid researchers in determining a early stopping criterion throughearly favorable treatment identification. This early stopping criterion will enable theavoidance of the ever-present ethical issues seen with unfavorable treatment assign-ment.A budget size of N = 100 was chosen and a sensitivity analysis was conducted usingvarious values for δ t , ω t , and c t B , while keeping Q t = 1. The values chosen for µ B were1 - 5, leading to H A : δ t = 1 through H A : δ t = 5. This lead to the hypothesis H : δ t = 0 H A : δ t (cid:54) = 0 (7)where δ t = µ A − µ B such that δ t = 1 , , , ,
5. Furthermore, ω t = 0 . , . , . c t B = 0 . , . , . ω t represents an increased cer-tainty of between time variability impact. Finally, decreasing the values of c t B results inan increased knowledge group B has no effect. The weighted allocation proportion val-6es in Figure 2 represent each of the δ t and ω t values. However only the c t B = 0 . N = 100 and retaining Q t = 1 throughout thesensitivity analysis the varied values of δ t represent 1% to a 5% difference in the twotreatments.The first analysis used δ t = 1 with ω t = 0 . ω t = 0 .
01 in Figure 2b the meanproportion of allocation values to treatment A decreased to 0.595, while treatmentB allocation increased to 0.405. However, the mean allocation switch from B to Aincreased slightly from 39.595 to 40.913. Finally, Figure 2c provides the results whenletting ω t = 0 . δ t = 1 and patient entrytime variances this accurate lead to the highest mean treatment allocation switch fromB to A, 46.702.Next δ t = 3 was chosen and the analysis was conducted. Using ω t = 0 . δ t = 1 to 18.217 using δ t = 3. When ω t = 0 .
01, seen in Figure 2e, the mean proportion of allocation values to treatmentA decreased to 0.751, while treatment B allocation increased to 0.249. However, themean allocation switch from B to A decreased from 40.913 at δ t = 1 to 24.694 using δ t = 3. Finally, Figure 2f shows the results when ω t = 0 . δ t = 1 to 39.499 using δ t = 3.Finally δ t = 5 was analyzed using the varied ω t values. Using ω t = 0 . ω t = 0 . δ t = 1 or 3. When ω t was decreased to 0.01, seen in in Figure 2h, the meanproportion of allocation values to treatment A decreased slightly to 0.833, while treat-ment B increased to 0.167. The mean number at which treatment allocation switchedfrom A to B was 15.453, almost double that obtained using ω t = 0 .
1. Lastly, Figure 2i.shows the allocation weights when the value for ω t was chosen to be 0.001. Here treat-ment A had a mean allocation proportion allocation of the mean proportion of 0.665,while treatment B had a mean allocation proportion of 0.335, seen in Figure 2o. Usingthe more precise patient entry time variances, treatment allocation switched from Bto A was 33.482, which is double the value at ω = 0 .
01 and 4 times that when ω = 0 . ω t within each δ t i , it can be seen the mean allocationprobabilities to treatment A decrease within each group, leading to lower convergentvalues in each δ t group, i.e. when δ t = 1 the mean convergent values for treatmentA go from 0.603, 0.595, 0.538 as information regarding ω t became more precise. Thisindicates a higher treatment B allocation proportion. However, increasing δ t also leadsto an increased mean number of necessary allocations for more precise ω t . For instance,when δ t = 3, the number of allocation values are 18.217, 24.694, and 39.499 for ω =0 . , . , .
001 respectively. However, when each of the allocation values are comparedwith comparable values of ω t at each δ t value, one may observe a diminished mean7umber for comparable values of ω t . For example, when allowing ω t = 0 .
1, the meannumber goes from 39.595 at δ t = 1 to 18.217 at δ t = 3 to 8.159 when δ t = 5. In fact,it appears using δ t = 5 provides the lowest mean switching value at every comparablevalue of ω t . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (a) δ t = 1 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (b) δ t = 1 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (c) δ t = 1 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (d) δ t = 3 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (e) δ t = 3 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (f) δ t = 3 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (g) δ t = 5 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (h) δ t = 5 , ω t = 0 . llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (i) δ t = 5 , ω t = 0 . Figure 2.
Comparison of Weight Allocation proportions for ω t = 0 . , .
01 and 0 .
001 and δ t = 1 , , C t B =0 . β t = 1 with bars representing the uncertainty across simulations.
2. Stopping Rule
In order to maintain a fully Bayesian approach to this research, a Bayes Factor wasused to determine definitive results. Additionally the 95% credible intervals and theassociated medians were calculated and were used, along with the Bayes Factor, todetermine when the algorithm flipped treatment assignment. In order to determinea “critical” Bayes Factor value, [31], suggest using a Bayes Factor greater than 100provides “Decisive evidence” against the null hypothesis of no difference. However,8he notation of [32] was used for the calculation of the Bayes Factor, whereby the nullhypothesis is in the numerator yielding p ( H | ( D )) = P ( D | H ) P ( H )) P ( D | H ) p ( H ) + P ( D | H ) p ( H ) (8)In their definition, they have the null hypothesis in the numerator and this leads tothe Bayes Factor BF = P ( D | H ) P ( D | H ) (9)Using equation 9, a Bayes Factor less than was chosen to provide “Decisiveevidence” and support towards the more favorable treatment.The Bayes Factor was calculated using the Bayesian Two Sample T-Test discussedin [32]. They define the Bayes Two Sample T Test as BF = T ν ( t | , T ν ( t | n δ λ, n δ σ δ ) (10)By choosing a Bayes Factor less than any Bayes Factor considered “Decisive”represented a 100 times more likely chance the allocation had switched. Any indecisiveBayes Factor indicated the budget size N = 100 was exhausted and no treatmentallocation switch had occurred. Parenthetical values in Table 3 and Table 4 representmedian and 95% credible interval values of the Bayes Factor while the bold numbersrepresent the Bayes Factor calculated at N = 100.Using the value c t B = 0 . δ t = 1 , ω t . These median, 95% credible intervals, and Bayes Factorsmay be seen in Table 3 and Table 4. Any italicized Bayes Factor is considered highlydecisive, and represents 100 times more likely a switch occurred. Table 3.
Covariate Included Budget Allocation N using δ t = 1 , , Q . , Q . , Q . ) P ( N ≥ ) δ t C t ω t (22, 32, 51), (22, 28, 45.025), (44, 55, 74), (48, 56, 69.025), (100,100,100), (76, 90, 100), (21, 31, 48), (23, 28, 46), (46, 58, 75), (49, 57, 68.025), (100,100,100), (98,100,100), (23, 32, 52), (22, 27.5, 43), (47, 57, 76), (49.975, 57, 69), (100, 100, 100), (99, 100, 100), Notice at ω t = 0 . δ t = 1 is 0.009 (median =47, 95% credibleinterval (26.975, 87)), while for δ t = 2 the Bayes Factor is 0.000 (median = 32, 95%credible interval (23, 52)). When analyzing δ t = 3, a Bayes Factor of 0.000 was calcu-lated (median = 27.5, 95% credible interval (22, 43)) An increase to δ t = 4 yielded a9ayes Factor of 0.010 (median = 28, 95% credible interval (23, 73.025)). Each of thesefirst 4 means indicated decisive evidence. However, when δ t = 5 the Bayes factor is0.088 (median = 32, 95% credible interval (25, 100)), indicating indecisive evidencesuggesting no switch to the better treatment occurred prior to exhausting the patientbudget size. Table 4.
Budget Allocation using N using µ B = 4 , Q . , Q . , Q . ) P ( N ≥ ) µ B C t ω t (25.000, 32.000,100.000), (45.000, 56.000, 85.000), (57.000, 68.000, 87.000), (26.000, 31.000, 100.000), (46.000, 55.000, 73.000), (78.000, 86.000, 99.000), (25.000, 32.000, 100.000), (45.000, 56.000, 76.025), (79.000, 87.000, 100.000), When ω t was reduced to 0.01, using δ t = 1 a Bayes Factor of 0.114 (median =74, 95% credible interval (50, 100)) was calculated indicating no decisive evidence ofpreferred treatment was found by N = 100. Yet, when δ t = 2 a Bayes Factor of 0.001(median = 57, 95% credible interval (47, 76)) indicated decisive evidence. Decisiveevidence was also seen when δ t = 3 with its Bayes Factor of 0.000 (median = 57,95% credible interval (49.975, 69)). Interestingly, using the value of δ t = 4 and δ t = 5yielded Decisive Bayes Factors (0.000 and 0.002 respectively), which provided highlydecisive evidence the allocation to the better treatment had occurred. However, using δ t = 4 median value was 61 (95% credible interval 49, 76.000), while with µ B = 5 alower median value of 56 was observed with 95% credible interval (45, 76.025).Lastly, when ω = 0 .
001 was analyzed, the Bayes Factor for δ t = 1 and δ t = 2 werethe same; a value of 1.000 which indicated no decisive evidence was found. Additionally,each had the same median and 95% credible interval values of 100. The Bayes Factordecreased to 0.974 (median = 100, 95% credible interval (99, 100)) when δ t = 3,however this was also indecisive. Likewise, even though the Bayes Factors decreaseddramatically when δ t = 4 , δ t = 4 median value was94 with 95% credible interval values (85, 100), however, when µ B = 5, median valuewas 87, with 95% credible interval values (79, 100).This analysis provides insight into how researchers may plan patient budget sizes.When choosing mean difference values between 1 and 3, it appears as though usingthe mean difference of 3 provides the lowest median and credible interval values usinglow to moderate belief in the variability between patients. However, it appears asthough δ t = 5 provides the lowest median value when using a moderate varianceof ω t = 0 .
01 using c t B = 0 . ω t = 0 . c t B = 0 . ω t = 0 .
01 showed Decisive evidence for all meanvalues except δ t = 1 10 lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (a) µ B = 1 , β t = 1. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (b) µ B = 1 , β t = 2. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (c) µ B = 3 , β t = 1. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (d) µ B = 3 , β t = 2. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (e) µ B = 5 , β t = 1. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (f) µ B = 5 , β t = 2. Figure 3.
Weight Allocation Proportion Comparisons when µ B = 1 , , ω t = 0 . c t B = 0 . β = 1 ,
3. Covariate Comparison
A comparison of the mean proportional allocation to treatments was also conductedwhen using values of β = 1 ,
2. This comparison was conducted using δ t = 1 , , ω t = 0 . c t B = 0 . β = 1 and β = 2 for eachof the δ t , ω t , c t B may be seen in Figure 3 and the results of the Bayes Factor may beseen in Table 5. For the graphs of δ t = 1 , , , , ω t = 0 . c t B = 0 . δ t = 1 and β t = 1 the mean proportion allocation for treatment A is0.608, while the mean proportion allocation for treatment B is 0.392. However, when β t is increased to 2, the mean proportion allocation for Treatment A shows only aslight decrease to 0.604 while the mean proportion allocation for treatment B is 0.396,illustrating when using δ t = 1 there is very little change in allocation proportion basedon the change in covariate value from 1 to 2. Likewise, the allocation switch from11 able 5. Budget Allocation using N using µ B = 4 , Q . , Q . , Q . ) P ( N ≥ ) for β t = 1 , β t δ t (26.000, 49.000, 88.025), (21.000, 28.000, 44 ), (26.000, 31.000, 100.000), B to A using β t = 1 was 39.813, with a similar value of 39.554 when using β t = 2,indicating changing the covariate value had little impact on when the switch from Bto A occurred.When using δ t = 3 and β t = 1 the mean proportion allocation for treatment A is0.791, while the mean proportion allocation for treatment B is 0.209. However, when β t is increased to 2, these allocation proportions remain the same illustrating when using δ t = 3 there is no change in allocation proportion when the covariate value increasesfrom 1 to 2. Likewise, the allocation switch from B to A using β t = 1 was 18.568,with a similar value of 18.544 when using β t = 2. This again indicates increasing thecovariate value from 1 to 2 has little to no impact on when the algorithm will switchfrom B to A.When using δ t = 5 and β t = 1 the mean proportion allocation for treatment Ais 0.888, while the mean proportion allocation for treatment B is 0.112. However,when β t is increased to 2, the mean proportion allocation for Treatment A shows onlya slight increase to 0.889 while the mean proportion allocation for treatment B isslightly decreased to 0.111, illustrating when using δ t = 5 there is very little change inallocation proportion when β t is increased from 1 to 2. Likewise, the allocation switchfrom B to A using β t = 1 was 8.359, with a similar value of 8.375 when using β t = 2.Similar to both δ = 1 and δ = 3 this indicates increasing the covariate value from 1to 2 has little to no impact on when the algorithm will switch from B to A.Finally, the median and 95% credible intervals and Bayes Factor were calculatedfor for δ = 1 , , β t = 1 , δ = 1 and 3, decisive Bayes factors were found at both β t = 1 and 2. The medianvalue for the combination δ t = 1 and β t = 1 was 50 (95% credible interval 27.000,86.025) with a Bayes factor of 0.008, while when β t = 2 the median was 49.000 (95%credible interval 26.000, 88.025) with a Bayes factor of 0.007. When δ t is increasedto 3, the median value decreases to 28 (95% credible interval 23.000, 43.025) with aBayes factor of 0.000 using β t = 1, yet when β t = 2, while the median value of 28remains unchanged, the 95% credible interval ranges from 21.000 to 44.000, and theBayes factor value of 0.000 remains unchanged. Finally, when δ t is increased to 5, amedian value of 32.000 is found using β t = 1 with 95% credible interval 26.000, 100.00and an indecisive Bayes factor of 0.083. When β t is increased to 2, the median valueonly slightly changes from 32 to 31, however, the 95% credible interval values remainunchanged (26.000, 100.000), again with an indecisive Bayes factor of 0.084. Conclusion
Researchers conducting Bayesian Random Allocation models for clinical trials can befaced with computationally intensive problems when running large scale simulations12equiring MCMC methods. These models are further complicated when a covariateis introduced. In the current application, a DLM was applied to random allocationmodels with a single covariate to demonstrate the ability to reduce time and patientallocation size in the presence of a covariate. Additionally, a sensitivity analysis wasconducted both on mean proportion of allocation to each treatment and mean valuerequired to switch to the preferred treatment. This provides insight for researchers whowish to know what treatment allocation proportion may be expected using varyingdifference values between µ A and µ B , between time variances ω t and current treatmentB variance c t B , thereby providing insight into the different model behaviors. Likewise,a power analysis was conducted using a Bayes Factor. This power analysis indicatedthe lowest median Bayes Factor occurred for a difference δ t = 5 using ω t = 0 . δ t = 1 , , ω t = 0 . c t B = 0 . δ t = 3 provides the lowestmedian and the most decisive Bayes factor, regardless of which β t is chosen. Likewise,it appears that increasing β t from 1 to 2 has little to no impact on model performance.Future works may include a sensitivity analysis using multiple values for β with largerdifferences than an increase of 1 unit. Additionally an examination of multi-arm studieswith covariates, and survival analysis applications should be studied.
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Am Stat . 2005;59(3):252257. . Appendix A. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (a) µ B = 1 , β t = 1 (b) µ B = 1 , β t = 2 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (c) µ B = 2 , β t = 1 (d) µ B = 2 , β t = 2 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (e) µ B = 3 , β t = 1 (f) µ B = 3 , β t = 2 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (g) µ B = 4 , β t = 1 (h) µ B = 4 , β t = 2 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . . samples P r opo r t i on s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll (i) µ B = 5 , β t = 1 (j) µ B = 5 , β t = 2 Figure 4.
Weight Allocation Proportion Comparisons when µ B = 1 , , , , ω t = 0 . c t B = 0 . β = 1 ,2 The bars represent the uncertainty across simulations