Statistical Methods for Composite Endpoints: Win Ratio and Beyond

Chapter 4 - Semiparametric Regression

Lu Mao

lmao@biostat.wisc.edu

Department of Biostatistics & Medical Informatics

University of Wisconsin-Madison

Aug 3, 2024

Outline

Proportional win-fractions (PW) model
- Model specification
  - An extension of Cox PH model and two-sample WR
- Estimation and residual analysis
- HF-ACTION example (WR package)
Generalized proportional odds model
- Model specification and interpretation
- Challenges in estimation
\[\newcommand{\d}{{\rm d}}\] \[\newcommand{\T}{{\rm T}}\] \[\newcommand{\dd}{{\rm d}}\] \[\newcommand{\cc}{{\rm c}}\] \[\newcommand{\pr}{{\rm pr}}\] \[\newcommand{\var}{{\rm var}}\] \[\newcommand{\se}{{\rm se}}\] \[\newcommand{\indep}{\perp \!\!\! \perp}\] \[\newcommand{\Pn}{n^{-1}\sum_{i=1}^n}\] \[ \newcommand\mymathop[1]{\mathop{\operatorname{#1}}} \] \[ \newcommand{\Ut}{{n \choose 2}^{-1}\sum_{i<j}\sum} \] \[ \def\a{{(a)}} \def\b{{(1-a)}} \def\t{{(1)}} \def\c{{(0)}} \def\d{{\rm d}} \def\T{{\rm T}} \def\bs{\boldsymbol} \]

Regression Framework

Regression vs two-sample
- Adjustment for confounding
- Quantitative predictors
- Statistical efficiency
- Screening of prognostic factors
  - Treatment, demographics (e.g., age, race, sex), medical history (e.g., diabetes, prior CVD), current medication (e.g., $\beta$-blocker, ACE inhibitor)

	Univariate	Hierarchical composite
Hypothesis testing	Gehan, log-rank	Win ratio/odds, net benefit (Ch 2)
Nonparametric estimation	RMST	RMT-IF (Ch 3)
Semiparametric regression	Cox PH model	?

PW Model

Win Ratio Regression

Modeling target
- Two independent subjects $(\mathcal H^*_i, Z_i)$ and $(\mathcal H^*_j, Z_j)$
  - $E\{\mathcal W(\mathcal H^*_i,\mathcal H^*_j)(t)\mid Z_i, Z_j\}$: Conditional win fraction (probability) for $i$ against $j$ at $t$
  - $E\{\mathcal W(\mathcal H^*_j,\mathcal H^*_i)(t)\mid Z_i, Z_j\}$: Conditional win fraction (probability) for $j$ against $i$ at $t$
- Covariate-specific win ratio \[\begin{equation}\label{eq:cov_spec_curtail_wr} WR(t; Z_i, Z_j;\mathcal W):= \frac{E\{\mathcal W(\mathcal H^*_i,\mathcal H^*_j)(t)\mid Z_i,Z_j\}}{E\{\mathcal W(\mathcal H^*_j,\mathcal H^*_i)(t)\mid Z_i, Z_j\}} \end{equation}\]
- Regress $WR(t; Z_i, Z_j;\mathcal W)$ vs $(Z_i, Z_j)$
  - $WR(t; Z_i, Z_j;\mathcal W) \geq 0$
  - $WR(t; Z_i, Z_j;\mathcal W) = WR(t; Z_j, Z_i;\mathcal W)^{-1}$

Model Specification

Proportional win-fractions (PW) model
- Multiplicative effects (Mao & Wang, 2021) \[\begin{equation}\label{eq:wr_reg} WR(t\mid Z_i, Z_j;\mathcal W)=\exp\left\{\beta^{\rm T}\left(Z_i- Z_j\right)\right\} \end{equation}\]
- PW: covariate-specific win/loss fractions proportional over time
  - WR constant over time
- $\beta$: log-WR associated with unit increases in covariates (regardless of follow-up time)
- Semiparametric: Parametric covariate effects, nonparametric otherwise
- Denote model by PW($\mathcal W$)
  - Stresses dependency on win function

Special Cases

Two-sample WR
- $\exp(\beta)$: WR comparing group $Z=1$ with group $Z=0$
Cox PH model
- PW$(\mathcal W_{\rm TFE})$ $\Leftrightarrow$ Cox PH model on TFE $\tilde T$ \[ \pr(t\leq \tilde T<t+\dd t\mid \tilde T\geq t; Z) =\exp(-\beta^\T Z)\lambda_0(t)\dd t \]
Bivariate Lehmann model
- PW$(\mathcal W_{\rm P})$ $\Leftarrow$ Lehmann model on $(D, T_1)$ \[ \pr(D>s, T_1 > t\mid Z) = H_0(s, t)^{\exp(-\beta^\T Z)} \]
  - PH on both $D$ and $\{T_1\mid D\}$ with same HRs $\exp(-\beta)$

Censored Data and Win Residual

Observed outcomes
- $\mathcal H^*(X_i)$: outcomes up to $X_i=D_i\wedge C_i$
- Observed win process $\delta_{ij}(t)=\mathcal W(\mathcal H^*_i,\mathcal H^*_j)(X_i\wedge X_j\wedge t)$
- Determinacy (win or loss) indicator: $R_{ij}(t)=\delta_{ij}(t)+\delta_{ji}(t)$
Win residuals
- Definition \[\begin{equation}\label{eq:wr:resid} M_{ij}(t\mid Z_i, Z_j;\beta)=\underbrace{\delta_{ij}(t)}_{\rm observed\,\,win} - \underbrace{R_{ij}(t)\frac{\exp\left\{\beta^{\rm T}\left( Z_i- Z_j\right)\right\}}{ 1+\exp\left\{\beta^{\rm T}\left(Z_i- Z_j\right)\right\}}}_{\rm model-based\,\, prediction} \end{equation}\]
- $E\{M_{ij}(t\mid Z_i, Z_j;\beta)\mid Z_i, Z_j\} =0$ for all $t$

Estimation and Inference

Estimating equation
- Sum of all possible pairs of covariate-/time-weighted residuals \[\begin{equation}\label{eq:wr:ee} \Ut\int_0^\infty (Z_i - Z_j) h(t; Z_i, Z_j)\dd M_{ij}(t \mid Z_i, Z_j;\beta)=0 \end{equation}\]
- Weight function $h(t; Z_i, Z_j)\equiv 1$
- Newton-Raphson to get root $\hat\beta$
- $\hat\var(\hat\beta)$: $U$-statistic variance

Checking Proportionality

Cumulative residuals
- Rescaled $\hat U_n(t)=\Ut(Z_i - Z_j)M_{ij}(t \mid Z_i, Z_j;\hat\beta)$
- Similar to score processes in Cox model (Lin et al., 1993)

Stratified Model

Address non-proportionality
- Categorical predictor $\to$ stratifier
Stratified PW model
- Idea: within-stratum comparisons (Dong et al., 2017, 2023)
- Model specification (Wang & Mao, 2022) \[ \frac{E\{\mathcal W(\mathcal H^*_{li},\mathcal H^*_{lj})(t)\mid Z_{li},Z_{lj}\}}{E\{\mathcal W(\mathcal H^*_{lj},\mathcal H^*_{li})(t)\mid Z_{li}, Z_{lj}\}} = \exp\left\{\beta^{\rm T}\left(Z_{li}- Z_{lj}\right)\right\} \]
  - $(H^*_{li}, Z_{li}), (H^*_{lj}, Z_{lj})$: $i$th and $j$th outcome-covariates in $l$th stratum $(l= 1,\ldots, L)$
  - Proportionality required only within stratum, not between

Software: `WR::pwreg()`

Basic syntax for PW$(\mathcal W_{\rm P})$
- (ID, time, status): same as WR::WRrec()
- Z: covariate matrix; strata: possible stratifier (categorical)

library(WR)
# fit PW model (death > nonfatal event)
obj <- pwreg(ID, time, status, Z, strata = NULL)

Output: an object of class pwreg
- obj$beta: $\hat\beta$
- obj$Var: $\hat\var(\hat\beta)$
- print(obj) to summarize regression results

Software: `WR::score.proc()`

Checking proportionality
- obj: a pwreg object

# compute covariate-specific cumulative residuals
score.obj <- score.proc(obj)

Output: an object of class score.proc
- score.obj$t: $t$
- score.obj$score: a matrix with rescaled residual process for each covariate per row
- plot(score.obj, k): plot the rescaled residuals for $k$th covariate

HF-ACTION: Non-Ischemic

Study information
- Population: 451 non-ischemic HF patients in HF-ACTION
- Outcome: death > (first) hospitalization
- Covariates: treatment, age, sex, race, BMI, LVEF, histories of hypertension, COPD, diabetes, current use of ACE inhibitor, $\beta$-blocker, smoking status

library(WR)
head(non_ischemic)
#>   ID time status trt_ab age sex Black.vs.White Other.vs.White   bmi
#> 1  1  221      2      0  62   1              0              0 25.18
#> 2  1  383      0      0  62   1              0              0 25.18
#> 3  2   23      2      0  75   1              1              0 22.96
#> 4  2 1400      0      0  75   1              1              0 22.96
#> ...

HF-ACTION: Table One

Table 1. Patient characteristics in non-ischemic cohort of HF-ACTION
	Usual care (N=231)	Training (N=220)
Age (years)	56 (46, 65.5)	54 (46, 62.2)
Sex - Female	153 (66.2%)	121 (55%)
Sex - Male	78 (33.8%)	99 (45%)
Race - White	117 (50.6%)	111 (50.5%)
Race - Black	103 (44.6%)	101 (45.9%)
Race - Other	11 (4.8%)	8 (3.6%)
BMI	31.3 (26.3, 37.2)	31 (25.8, 36.5)
LVEF (%)	25.1 (20.9, 31.3)	25.1 (20.9, 31.2)
Hypertension	129 (55.8%)	129 (58.6%)
COPD	21 (9.1%)	15 (6.8%)
Diabetes	71 (30.7%)	58 (26.4%)
ACE Inhibitor	174 (75.3%)	167 (75.9%)
Beta Blocker	223 (96.5%)	211 (95.9%)

HF-ACTION: PW Regression

Fit PW($\mathcal W_{\rm P}$)
- Death > (first) hospitalization

# number of covariates (-c(ID, time, status))
p <- ncol(non_ischemic) - 3

# extract ID, time, status and covariates matrix Z from the data.
# note that: ID, time and status should be column vector
ID <- non_ischemic[,"ID"]
time <- non_ischemic[,"time"] / 30.5 # days to months
status <- non_ischemic[,"status"]
Z <- as.matrix(non_ischemic[, 4:(3+p)])

# pass the parameters into the function
obj <- pwreg(ID, time, status, Z)

HF-ACTION: Results (I)

Model summary

obj
#> Call:
#> pwreg(ID = ID, time = time, status = status, Z = Z)

#> Total number of pairs: 101475 
#> Wins-losses on death:  7644 (7.5%) 
#> Wins-losses on non-fatal event:  78387 (77.2%) 
#> Indeterminate pairs 15444 (15.2%) 
#> 
#> Newton-Raphson algorithm converged in 5 iterations.
#> 
#> Overall test: chisq test with 13 degrees of freedom; 
#>  Wald statistic 24.9 with p-value 0.02392931

HF-ACTION: Results (II)

Inference table for $\hat\beta$
- Age, black vs white, LVEF significant

obj
#>                     Estimate         se z.value p.value  
#> Training vs Usual  0.1906687  0.1264658  1.5077 0.13164  
#> Age (year)        -0.0128306  0.0057285 -2.2398 0.02510 *
#> Male vs Female    -0.1552923  0.1294198 -1.1999 0.23017  
#> Black vs White    -0.3026335  0.1461330 -2.0709 0.03836 *
#> Other vs White    -0.3565390  0.3424360 -1.0412 0.29779  
#> BMI               -0.0181310  0.0097582 -1.8580 0.06316 .
#> LVEF               0.0214905  0.0086449  2.4859 0.01292 *
#> Hypertension      -0.0318291  0.1456217 -0.2186 0.82698  
#> COPD              -0.4023069  0.2066821 -1.9465 0.05159 .
#> Diabetes           0.0703990  0.1419998  0.4958 0.62006  
#> ACE Inhibitor     -0.1068201  0.1571317 -0.6798 0.49662  
#> Beta Blocker      -0.5344979  0.3289319 -1.6250 0.10417  
#> Smoker            -0.0602350  0.1682826 -0.3579 0.72039

HF-ACTION: Race Effect

Joint test $(\chi_2^2)$ on race categories
- Black/African American, white, other

# extract estimates of (beta_4, beta_5)
beta <- matrix(obj$beta[4:5])
# extract estimated covariance matrix for (beta_4, beta_5)
Sigma <- obj$Var[4:5, 4:5]
# compute chisq statistic in quadratic form
chistats <- t(beta) %*% solve(Sigma) %*% beta  

# compare the Wald statistic with the reference
# distribution of chisq(2) to obtain the p-value
1 - pchisq(chistats, df = 2)
#>           [,1]
#> [1,] 0.1016988

HF-ACTION: Win Ratio Table

Covariate-specific WRs $\exp(\hat\beta)$
- Training wins 21.0% more than usual care

#> Point and interval estimates for the win ratios:
#>                   Win Ratio 95% lower CL 95% higher CL
#> Training vs Usual 1.2100585    0.9444056     1.5504374
#> Age (year)        0.9872513    0.9762288     0.9983983
#> Male vs Female    0.8561648    0.6643471     1.1033663
#> Black vs White    0.7388699    0.5548548     0.9839127
#> Other vs White    0.7000951    0.3578286     1.3697431
#> BMI               0.9820323    0.9634287     1.0009952
#> LVEF              1.0217231    1.0045572     1.0391823
#> Hypertension      0.9686721    0.7281543     1.2886357
#> COPD              0.6687755    0.4460178     1.0027865
#> Diabetes          1.0729362    0.8122757     1.4172433
#> ACE Inhibitor     0.8986873    0.6604773     1.2228110
#> Beta Blocker      0.5859634    0.3075270     1.1164977
#> Smoker            0.9415433    0.6770144     1.3094312

HF-ACTION: Residual Analysis

Check proportionality assumption on covariates

# compute score processes
score_obj <- score.proc(obj)
# plot scores for first 8 covariates
par(mfrow = c(2, 4))
for(i in c(1 : 8)){
  plot(score_obj, k = i, xlab = "Time (months)")
  # add reference lines
  abline(a = 2, b = 0, lty = 3)
  abline(a = 0, b = 0, lty = 3)
  abline(a = - 2, b = 0, lty = 3)
}

HF-ACTION: Residual Plot

Mostly well-behaved
- Exercise: Fit stratified PW model by sex

Generalized PO Model

Alternative Modeling

Extension of proportional odds (PO)?
- Univariate PO model (Bennett, 1983) \[ \log\left\{\frac{\pr(D\leq t)}{\pr(D > t)}\right\}=h(t) + \beta^\T Z \]
  - $\exp(\beta)$: Odds ratios (OR) of having early death (regardless of time cut-off)

Hypothesis testing

Gehan, log-rank

Win ratio/odds, net benefit (Ch 2)

Nonparametric estimation

RMST

RMT-IF (Ch 3)

Semiparametric regression

Cox PH model

PO model

PW model (Ch 4)

Model Specification

Outcome formulation
- Multistate process $Y(0)=0, 1,\ldots, K, \infty$ (hierarchically ordered)
Generalized PO model \[ \log\left[\frac{\pr\{Y(t)>k\}}{\pr\{Y(t)\leq k\}}\right]=h_k(t) + \beta^\T Z \]
- $\exp(\beta)$: ORs of having early and less favorable outcomes (regardless of time and favorability cut-off)
  - Two-way (time & order) proportionality
  - $K=0$ $\to$ Univariate PO

Estimation Problem

Main challenge
- Nonparametric baseline log-odds $h_k(t)$
- Model does not determine likelihood $\to$ ~~MLE~~
  - Unlike univariate PO (Murphy et al., 1997)
Possible solutions
- Posit a nuisance (Markov) model for $Y(t)$ $\to$ MLE
- Find estimating functions under marginal model alone

Conclusion

Notes

More on PW
- Check functional form (linear, quadratic, grouped) of $Z$ (Lin et al., 1993)
  - E.g., is age effect linear?
  \[ \mbox{plot} \sum_{i, j: Z_i - Z_j\leq z}(Z_i - Z_j)M_{ij}(\infty \mid Z_i, Z_j;\hat\beta) \mbox{ against } z\in\mathbb R \]
- Currently WR::pwreg() implements only PW($\mathcal W_{\rm P}$)
  - More functionalities to be added
- Variations: regression of win ratio/odds (Follmann et al., 2019; Song et al., 2022)
Refinement of PO
- $\beta^\T Z \to \beta^\T Z_{(1)} + \gamma_k^\T Z_{(2)}$: relaxes order-proportionality on $Z_{(2)}$

Summary

PW: WR regression
- Proportionality and multiplicativity \[ WR(t\mid Z_i, Z_j;\mathcal W)=\exp\left\{\beta^{\rm T}\left(Z_i- Z_j\right)\right\} \]
  - $\exp(\beta)$: WRs with unit increases in covariates
  - WR::pwreg(ID, time, status, Z, strata)
Generalize PO model
- Multistate process $Y(t)=0, 1, \ldots, K, \infty$ \[ \log\left[\frac{\pr\{Y(t)>k\}}{\pr\{Y(t)\leq k\}}\right]=h_k(t) + \beta^\T Z \]
  - $\exp(\beta)$: ORs with unit increases in covariates

References

Bennett, S. (1983). Analysis of survival data by the proportional odds model. Statistics in Medicine, 2(2), 273–277. https://doi.org/10.1002/sim.4780020223

Dong, G., Hoaglin, D. C., Huang, B., Cui, Y., Wang, D., Cheng, Y., & Gamalo-Siebers, M. (2023). The stratified win statistics (win ratio, win odds, and net benefit). Pharmaceutical Statistics, 22(4), 748–756. https://doi.org/10.1002/pst.2293

Dong, G., Qiu, J., Wang, D., & Vandemeulebroecke, M. (2017). The stratified win ratio. Journal of Biopharmaceutical Statistics, 28(4), 778–796. https://doi.org/10.1080/10543406.2017.1397007

Follmann, D., Fay, M. P., Hamasaki, T., & Evans, S. (2019). Analysis of ordered composite endpoints. Statistics in Medicine, 39(5), 602–616. https://doi.org/10.1002/sim.8431

Lin, D. Y., Wei, L. J., & Ying, Z. (1993). Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika, 80(3), 557–572. https://doi.org/10.1093/biomet/80.3.557

Mao, L., & Wang, T. (2021). A class of proportional win-fractions regression models for composite outcomes. Biometrics, 77(4), 1265–1275. https://doi.org/10.1111/biom.13382

Murphy, S. A., Rossini, A. J., & Vaart, A. W. van der. (1997). Maximum Likelihood Estimation in the Proportional Odds Model. Journal of the American Statistical Association, 92(439), 968–976. https://doi.org/10.1080/01621459.1997.10474051

Song, J., Verbeeck, J., Huang, B., Hoaglin, D. C., Gamalo-Siebers, M., Seifu, Y., Wang, D., Cooner, F., & Dong, G. (2022). The win odds: statistical inference and regression. Journal of Biopharmaceutical Statistics, 33(2), 140–150. https://doi.org/10.1080/10543406.2022.2089156

Wang, T., & Mao, L. (2022). Stratified proportional win-fractions regression analysis. Statistics in Medicine, 41(26), 5305–5318. https://doi.org/10.1002/sim.9570

Statistical Methods for Composite Endpoints: Win Ratio and Beyond

Outline

Regression Framework

PW Model

Win Ratio Regression

Model Specification

Special Cases

Censored Data and Win Residual

Estimation and Inference

Checking Proportionality

Stratified Model

Software: WR::pwreg()

Software: WR::score.proc()

HF-ACTION: Non-Ischemic

HF-ACTION: Table One

HF-ACTION: PW Regression

HF-ACTION: Results (I)

HF-ACTION: Results (II)

HF-ACTION: Race Effect

HF-ACTION: Win Ratio Table

HF-ACTION: Residual Analysis

HF-ACTION: Residual Plot

Generalized PO Model

Alternative Modeling

Model Specification

Estimation Problem

Conclusion

Notes

Summary

References

Software: `WR::pwreg()`

Software: `WR::score.proc()`