Win ratio regression analysis

Fit a multiplicative win-ratio regression model to partially ordered response against covariates.

Usage

wreg(Y, Z, fun = NULL, sfun = NULL, ep = 1e-06)

Arguments

Y

An \(n\times K\) matrix for \(K\)-variate response data on \(n\) subjects. The entries must be numeric. For pseudo-efficient estimation (without specifying sfun), the average score across components (row means) should be compatible with the partial order (i.e., preserve the same order for any two comparable and ordered elements).

Z

An \(n\times p\) design matrix for covariates.

fun

User-specified win function for pairwise comparison. It takes two arguments \(y_1\) and \(y_0\) (both \(K\)-vectors) and returns 1 if \(y_1\) wins, -1 if \(y_0\) wins, and 0 if tied. The default is wprod for the product order of multivariate ordinal data.

sfun

The scoring function used in pseudo-efficient estimation. The default is to take the row means of Y.

ep

Convergence criterion in Newton-Raphson algorithm. The default is 1e-6.

Value

An object of class wreg with the following components:

beta

A vector of estimated regression coefficients.

var

Estimated covariance matrix for beta

l

Number of Newton-Raphson iterations.

beta_nv

Naive (non-pseudo-efficient) estimates of beta.

se_nv

Estimated standard errors for beta_nv.

n

Sample size \(n\) of input data with non-missing values.

Nwl

Number of comparable pairs (those with a win and loss) out of the \(n(n-1)/2\) possible ones.

References

Mao, L. (2024). Win ratio for partially ordered data. Statistica Sinica, Under revision.

See also

wprod, print.wreg, summary.wreg.

Examples

head(liver)
#>   R1NASH R2NASH Sex    AF Steatosis  SSF2  LSN
#> 1      3      2   M FALSE        30  0.21 2.33
#> 2      1      1   F FALSE         5  0.38 2.86
#> 3      4      2   M FALSE        70  0.58 3.65
#> 4      4      4   F  TRUE        30 -0.08 2.73
#> 5      4      3   M  TRUE        70 -0.04 2.53
#> 6      3      3   M FALSE        10  0.02 2.88
# regress bivariate ratings against covariates
Y <- 5 - liver[, c("R1NASH", "R2NASH")] # lower score is better
Z <- cbind("Female" = liver$Sex == "F",
           liver[, c("AF", "Steatosis",   "SSF2",  "LSN")]) # covariates
obj <- wreg(Y, Z) # fit model
obj
#> Call:
#> wreg(Y = Y, Z = Z)
#> 
#> n = 154 subjects with complete data
#> Comparable (win/loss) pairs: 9548/11781 = 81%
#> 
#>    Female         AF   Steatosis         SSF2        LSN
#>  -0.18956 -0.9660827 -0.02779146 -0.007926333 -0.1029914
summary(obj)
#> Call:
#> wreg(Y = Y, Z = Z)
#> 
#> n = 154 subjects with complete data
#> Comparable (win/loss) pairs: 9548/11781 = 81%
#> 
#> Newton-Raphson algoritm converged in 7 iterations
#> 
#>                coef exp(coef)  se(coef)      z Pr(>|z|)    
#> Female    -0.189560    0.8273  0.259988 -0.729 0.465934    
#> AF        -0.966083    0.3806  0.280313 -3.446 0.000568 ***
#> Steatosis -0.027791    0.9726  0.005281 -5.262 1.42e-07 ***
#> SSF2      -0.007926    0.9921  0.003953 -2.005 0.044953 *  
#> LSN       -0.102991    0.9021  0.125718 -0.819 0.412657    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#>           exp(coef) exp(-coef) lower .95 upper .95
#> Female      0.82732    1.20872   0.49702    1.3771
#> AF          0.38057    2.62763   0.21970    0.6592
#> Steatosis   0.97259    1.02818   0.96258    0.9827
#> SSF2        0.99210    1.00796   0.98445    0.9998
#> LSN         0.90213    1.10848   0.70512    1.1542
#> 
#> Overall Wald test = 79.129 on 5 df,  p = 1.221245e-15