Statistical Inference of Covariate-Adjusted Randomized Experiments

Transcription

1 1 Statistical Inference of Covariate-Adjusted Randomized Experiments Feifang Hu Department of Statistics George Washington University Joint research with Wei Ma, Yichen Qin and Yang Li Nov 8, 2018 at IMA,

2 2 Outline Introduction Framework General Properties Implementation and Correction Numerical Studies Conclusion

3 1 Introduction 3 1 Introduction Covariate-adjusted randomization is frequently used because it utilizes the covariate information to form more balanced treatment groups. Balance categorical covariates: Pocock and Simon s minimization method and its extensions (Taves,1974; Pocock and Simon 1975; Hu and Hu 2012) Balance continuous covariates based on distribution characteristics, e.g., mean and variance (Frane 1998), quartiles (Su 2011), density function (Ma and Hu 2013). Balance continuous covariates based on models (Atkinson 1982, Smith 1984ab) Balance covariates available prior to the experiment onset (Morgan and Rubin, 2012, 2015, Qin et al. 2017)

4 1 Introduction 4 Since covariate-adjusted randomizations inevitably use the covariate information in forming more balanced treatment groups, the subsequent statistical inference is usually affected and demonstrates undesirable properties, such as reduced type I errors and powers. This phenomenon of conservativeness is particularly common for a working model including only a subset of covariates used in randomization, such as two sample t test.

5 1 Introduction 5 It is ideal that the covariates used in randomization should be included in the subsequent analysis to achieve valid test. However, unadjusted tests still dominate in practice (Sverdlov, 2015). Investigation sites Simplicity of the test procedure Robustness to model misspecification As covariates are commonly used in comparative studies (biomarker analysis, precision medicine and crowdsourced-internet experimentation), understanding the impact of covariate-adjusted randomization on statistical inference is an increasingly pressing problem.

6 1 Introduction 6 Existing work Birkett (1985), Forsythe (1987), etc.. mainly based on simulations. Shao et al. (2010) shows t-test is conservative for stratified biased coin design. Ma et al. (2015) studied tests under a linear model for discrete covariate-adjusted randomization by assuming that overall and marginal imbalances are bounded in probability.

7 1 Introduction 7 Limitations Not applicable to randomizations directly balancing continuous covariates, e.g., Atkinson s D A -Biased Coin Design. The assumed balancing properties are too strong, i.e., O p (1) marginal imbalances. Do not consider the scenario when covariate information are avialable before the experiment starts, e.g., Rerandomization, Pairwise Sequential Randomization.

8 1 Introduction 8 Motivations Derive the statistical properties of inference under general covariate-adjusted randomization methods. Explicitly display the relationship between covariate balance and inference, and explain why inference behaves differently for various randomization methods. Obtain the results that have broad applications, including RR, PSR, and D A -BCD, and compare these methods analytically. Propose a method to attain valid and powerful tests.

9 2 Framework 9 2 Framework Suppose that n units are to be assigned to two treatment groups. T i denotes the assignment of the i-th unit, i.e., T i = 1 for treatment 1 and T i = 0 for treatment 2. Let x i = (x i,1,..., x i,p+q ) t represent p + q iid covariates observed for the i-th unit, where x i,j X j for i = 1,..., n. The underlying model: Y i = µ 1 T i + µ 2 (1 T i ) + p+q j=1 β j x i,j + ɛ i, where µ 1 µ 2 is the treatment effect, β = (β 1,..., β p+q ) t is the covariate effects, and ɛ i is iid random error with mean zero and variance σ 2 ɛ, and is independent of covariates. Covariates are assumed independent of each other with EX j = 0 for j = 1,..., p + q.

10 2 Framework 10 After allocating the units to treatment groups via covariate-adjusted randomization, a working model is used to estimate and test the treatment effect. In such a working model, it is common in practice to include a subset of covariates used in randomization, or sometimes even no covariates at all (Shao et al. 2010, Ma et al. 2015, Sverdlov 2015). The working model:. E[Y i ] = µ 1 T i + µ 2 (1 T i ) + p β j x i,j. j=1

11 2 Framework 11 Let Y = (Y 1,..., Y n ) t, T = (T 1,..., T n ) t, X = [X in ; X ex ], where x 1,1 x 1,p x 1,p+1 x 1,p+q X in =....., X ex =..... x n,1 x n,p x n,p+1 x n,p+q. Further let β in = (β 1,..., β p ) t, β ex = (β p+1,..., β p+q ) t, so that β = (β t in, βt ex) t. Then the working model can also be written as, E[Y ] = Gθ, where G = [T ; 1 n T ; X in ] is the design matrix, θ = (µ 1, µ 2, β t in )t is the vector of parameters of interest, and 1 n is the n-dimensional vector of ones. The ordinary least squares (OLS) estimate of θ, ˆθ = (ˆµ 1, ˆµ 2, ˆβ t in )t is, ˆθ = (G t G) 1 G t Y.

12 2 Framework 12 Testing the treatment effect: and the test statistic is H 0 : µ 1 µ 2 = 0 versus H 1 : µ 1 µ 2 0, S = L t ˆθ ˆσ 2 w L t (G t G) 1 L, where L = (1, 1, 0,..., 0) t is a vector of length p + 2, and ˆσ 2 w = Y G ˆθ 2 /(n p 2) is the model-based estimate of the error variance σ 2 w = σ 2 ɛ + q j=1 β2 p+j Var(X p+j). The traditional testing procedure is to reject the null hypothesis at the significance level α if S > z 1 α/2, where z 1 α/2 is (1 α/2)-th quantile of a standard normal distribution.

13 2 Framework 13 Testing the covariate effects: Let C be an m (p + 2) matrix of rank m (m p) with entries in the first two columns all equal to zero (no treatment effect to test). and the test statistic is, H 0 : Cθ = c 0 versus H 1 : Cθ = c 1, (1) S = (C ˆθ c 0 ) t [C(G t G) 1 C t ] 1 (C ˆθ c 0 ) mˆσ 2 w The traditional testing procedure is to reject the null hypothesis at the significance level α if S > z 1 α/2, where z 1 α/2 is (1 α/2)-th quantile of a standard normal distribution.

14 3 General Properties 14 3 General Properties Assumption 1 Global balance: n 1 n i=1 (2T i 1) p 0. Assumption 2 Covariate balance: n 1/2 n i=1 (2T i 1) x i d ξ, where ξ is a (p+q)-dimensional random vector with E[ξ] = 0.

15 3 General Properties 15 Consistency: Theorem 3.1 Given Assumptions 1 and 2, we have ˆθ p θ.

16 3 General Properties 16 Testing the treatment effect: We partition ξ = (ξ t in, ξt ex) t so that ξ in represents the first p dimensions of ξ, and ξ ex the last q dimensions. Further let λ 1 = σ ɛ /σ w, λ 2 = 1/σ w, and Z be a standard normal random variable that is independent of ξ ex. Theorem 3.2 Given Assumptions 1 and 2, we have 1. Under H 0 : µ 1 µ 2 = 0, then S d λ 1 Z + λ 2 β t exξ ex. 2. Under H 1 : µ 1 µ 2 0, consider a sequence of local alternatives with µ 1 µ 2 = δ/ n for a fixed δ 0, then S d λ 1 Z + λ 2 β t exξ ex λ 2δ.

17 3 General Properties 17 The asymptotic distribution of test statistic S under H 0 consists of two independent components, λ 1 Z and λ 2 β t exξ ex. The first component is due to the random error ɛ i in the underlying model, and remains invariant under different covariate-adjusted randomization. The second component of S represents the impact of a covariate-adjusted randomization on the test statistic through the level of covariate balance. Under covariate-adjusted randomization, ξ is more concentrated around 0 as opposed to complete randomization, leading to conservative tests.

18 3 General Properties 18 Testing the covariate effects: Theorem 3.3 Given Assumptions 1 and 2, we have 1. Under H 0 : Cθ = c 0, then S d χ 2 m /m. 2. Under H 1 : Cθ = c 1, consider a sequence of local alternatives with c 1 c 0 = / n for a fixed 0, then S d χ 2 m (φ)/m, φ = t [CV 1 C t ] 1 /σ 2 w. where φ is the non-central parameter, and V = diag (1/2, 1/2, Var(X 1 ),..., Var(X p )).

19 3 General Properties 19 The type I error is maintained when testing the covariate effects under covariate-adjusted randomization. The power, however, is reduced if not all covariate information is incorporated in the working model.

20 4 Implementation and Correction 20 4 Implementation and Correction 4.1 Examples Complete Randomization Rerandomization (Morgan and Rubin, 2012, 2015) Repeat the traditional randomization process until a satisfactory configuration is achieved. Pairwise Sequential Randomization (Qin et al, 2017) An alternative that achieves the optimal covariate balance and is computationally more efficient. Atkinson s D A -Biased Coin Design (Atkinson 1982, Smith 1984ab) Represent a large class of methods that take covariates into account in allocation rules based on certain optimality criteria.

21 4 Implementation and Correction 21 Rerandomization (1) Collect covariate data. (2) Specify a balance criterion to determine when a randomization is acceptable. For example, the criterion could be defined as a threshold of a > 0 on some user-defined imbalance measure, denoted as M. (3) Randomize the units into treatment groups using traditional randomization methods, such as CR. (4) Check the balance criterion M < a. If the criterion is satisfied, go to Step (5); otherwise, return to Step (3). (5) Perform the experiment using the final randomization obtained in Step (4).

22 4 Implementation and Correction 22 Pairwise Sequential Randomization (1) Collect covariate data. (2) Choose the covariate imbalance measure for n units, denoted as M(n). (3) Randomly arrange all n units in a sequence x 1,..., x n. (4) Separately assign the first two units to treatment 1 and treatment 2.

23 4 Implementation and Correction 23 (5) Suppose that 2i units have been assigned to treatment groups (i 1), for the (2i + 1)-th and (2i + 2)-th units: (5a) If the (2i + 1)-th unit is assigned to treatment 1 and the (2i + 2)-th unit is assigned to treatment 2 (i.e., T 2i+1 = 1 and T 2i+2 = 0), then we can calculate the potential imbalance measure, M (1) i, between the updated treatment groups with 2i + 2 units. (5b) Similarly, if the (2i + 1)-th unit is assigned to treatment 2 and the (2i + 2)-th unit is assigned to treatment 1 (i.e., T 2i+1 = 0 and T 2i+2 = 1), then we can calculate the potential imbalance measure, M (2) i, between the updated treatment groups with 2i + 2 units.

24 4 Implementation and Correction 24 (6) Assign the (2i + 1)-th and (2i + 2)-th units to treatment groups according to the following probabilities: ρ if M (1) i < M (2) i P(T 2i+1 = 1 x 2i,..., x 1, T 2i,..., T 1 ) = 1 ρ if M (1) i > M (2), i 0.5 if M (1) i = M (2) i where 0.5 < ρ < 1, and assign T 2i+2 = 1 T 2i+1 to maintain the equal proportions. (7) Repeat Steps (5) through (7) until all units are assigned.

25 4 Implementation and Correction 25 Atkinson s D A -Biased Coin Design Suppose n units have been assigned to treatment groups, D A -BCD assigns the (n + 1)-th unit to treatment 1 with probability P(T n+1 = 1 x n+1,..., x 1, T n,..., T 1 ) = [1 (1; x t n+1)(f t nf n ) 1 b n ] 2 [1 (1; x t n+1 )(Ft nf n ) 1 b n ] 2 + [1 + (1; x t n+1 )(Ft nf n ) 1 b n ] 2. where F n = [1 n ; X] and b t n = (2T 1 n ) t F n.

26 4 Implementation and Correction 26 Complete Randomization ξ CR N(0, Σ) Rerandomization ξ RR Σ 1/2 D D t D < a Pairwise Sequential Randomization Atkinson s D A -Biased Coin Design ξ PSR = O p ( 1 n ) ξ D-BCD N(0, 1 5 Σ) where Σ = diag(var(x 1 ),..., Var(X p+q )), D N(0, I p+q ) and I p+q is the (p + q)-dim identity matrix.

27 4 Implementation and Correction 27 Testing the Treatment Effect under Atkinson s D A -Biased Coin Design Theorem 4.1 Under D A -BCD, we have 1. Under H 0 : µ 1 µ 2 = 0, then ( S d N 0, σ2 ɛ + 1 q 5 j=1 β2 p+j Var(X ) p+j) σɛ 2 + q. j=1 β2 p+j Var(X p+j) 2. Under H 1 : µ 1 µ 2 0, where µ 1 µ 2 = δ/ n for a fixed δ 0, ( S d 1 N 2 λ 2δ, σ2 ɛ + 1 q 5 j=1 β2 p+j Var(X ) p+j) σɛ 2 + q. j=1 β2 p+j Var(X p+j)

28 4 Implementation and Correction 28 Testing the Treatment Effect under Pairwise Sequential Randomization Theorem 4.2 Under PSR, we have 1. Under H 0 : µ 1 µ 2 = 0, then ( S d N 0, σ 2 ɛ σ 2 ɛ + q j=1 β2 p+j Var(X p+j) 2. Under H 1 : µ 1 µ 2 0, where µ 1 µ 2 = δ/ n for a fixed δ 0, ( ) S d 1 N 2 λ σɛ 2 2δ, σɛ 2 + q j=1 β2 p+j Var(X. p+j) ).

29 4 Implementation and Correction 29 The variance from the covariates is completely eliminated out in the numerator of the asymptotic distribution of S, resulting in a distribution more concentrated around 0 than the standard normal distribution. This can be considered as an extension of the results in Ma et al. (2015) that studied conservative tests for covariate-adaptive designs balancing discrete covariates.

30 4 Implementation and Correction Correction for Conservativeness To correct conservativeness, we need to obtain the correct asymptotic critical values for valid tests. Based on the asymptotic distribution of S in Theorem 3.2. Need to estimate the unknown parameters. Or use Bootstrap method to do the correction. Computationally intensive.

31 4 Implementation and Correction 31 Table 1: Comparison of different covariate-adjusted randomization procedures in terms of covariate balance, traditional tests conservativeness, and corrected tests powers.

32 5 Numerical Studies 32 5 Numerical Studies Verification of Theoretical Results Underlying model: Y i = µ 1 T i + µ 2 (1 T i ) + 4 β j x i,j + ɛ i, j=1 where µ 1 = µ 2 = 0, β j = 1 for j = 1,..., 4. x i,j N(0, 1) for j = 1,..., 4 and is independent of each other. The random error ɛ i N(0, 2 2 ) is independent of all x i,j. Working model:. E[Y i ] = µ 1 T i + µ 2 (1 T i ) + β 1 x i,1 + β 2 x i,2

33 5 Numerical Studies 33 Verification of Theoretical Results CR Rerandomization Atkinson PSR pdf Simulated Theoretical N(0,1) pdf Simulated Theoretical N(0,1) pdf Simulated Theoretical N(0,1) pdf Simulated Theoretical N(0,1) t t t t Figure 1: Comparison of theoretical distributions and simulated distributions of S. In each panel, red solid curve represents the simulated distribution, blue dash curve represents the theoretical distribution, and the gray bold curve is the standard normal density.

34 5 Numerical Studies 34 Conservative Hypothesis Testing for Treatment Effect Underlying model: Y i = µ 1 T i + µ 2 (1 T i ) + 6 β j x i,j + ɛ i, (2) j=1 where β j = 1 for j = 1,...6. x i,j N(0, 1) and is independent of each other. The random error ɛ i N(0, 2 2 ) is independent of all x i,j. Working model: W1: E[Y i ] = µ 1 T i + µ 2 (1 T i ). W2: E[Y i ] = µ 1 T i + µ 2 (1 T i ) + 2 j=1 β jx i,j. W3: E[Y i ] = µ 1 T i + µ 2 (1 T i ) + 6 j=3 β jx i,j. W4: E[Y i ] = µ 1 T i + µ 2 (1 T i ) + 6 j=1 β jx i,j.

35 5 Numerical Studies 35 Conservative Hypothesis Testing for Treatment Effect: Type I error Randomization W1 W2 W3 W4 CR RR D A -BCD PSR Table 2: Type I error of traditional tests for treatment effect using different working models and different randomization procedures.

36 5 Numerical Studies 36 Corrected Hypothesis Testing for Treatment Effect: Type I error Randomization W1 W2 W3 W4 CR RR D A -BCD PSR Table 3: Type I error of hypothesis testing for treatment effect using estimated asymptotic distribution s critical values under different working models and different randomization procedures.

37 5 Numerical Studies 37 Corrected Hypothesis Testing for Treatment Effect: Power CR Rerandomization Atkinson PSR Power W4 W3 W2 W1 Power W4 W3 W2 W1 Power W4 W3 W2 W1 Power W4 W3 W2 W u0 u1 u0 u1 u0 u1 u0 u1 Figure 2: Power against µ 1 µ 2 using estimated asymptotic distribution s critical values and p-values. Sample size n = 500. Note that we plot the power of W4 under CR in bold gray curves in all the panels for a better comparison among different randomizations.

38 6 Conclusion 38 6 Conclusion Derive inference properties under general covariate-adjusted randomization. Explicitly unveil the relationship between covariate-adjusted and inference properties. Apply the general theory to several important randomization methods. A correction approach is proposed to attain valid and powerful test.

39 6 Conclusion 39 Thank you!