Multiple imputation pooling: univariate version

Pools univariate estimates of m repeated complete data analysis

pool.scalar(Q, U, n = Inf, k = 1)

Arguments

Q	A vector of univariate estimates of `m` repeated complete data analyses.
U	A vector containing the corresponding `m` variances of the univariate estimates.
n	A number providing the sample size. If nothing is specified, an infinite sample `n = Inf` is assumed.
k	A number indicating the number of parameters to be estimated. By default, `k = 1` is assumed.

Value

Returns a list with components.

m:: Number of imputations.
qhat:: The m univariate estimates of repeated complete-data analyses.
u:: The corresponding m variances of the univariate estimates.
qbar:: The pooled univariate estimate, formula (3.1.2) Rubin (1987).
ubar:: The mean of the variances (i.e. the pooled within-imputation variance), formula (3.1.3) Rubin (1987).
b:: The between-imputation variance, formula (3.1.4) Rubin (1987).
t:: The total variance of the pooled estimated, formula (3.1.5) Rubin (1987).
r:: The relative increase in variance due to nonresponse, formula (3.1.7) Rubin (1987).
df:: The degrees of freedom for t reference distribution by the method of Barnard-Rubin (1999).
fmi:: The fraction missing information due to nonresponse, formula (3.1.10) Rubin (1987).

Details

The function averages the univariate estimates of the complete data model, computes the total variance over the repeated analyses, and computes the relative increase in variance due to nonresponse and the fraction of missing information.

References

Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley and Sons.

Author

Karin Groothuis-Oudshoorn and Stef van Buuren, 2009

Examples

# example with manual and automatic pooling
imp <- mice(nhanes, maxit = 2, m = 2, print = FALSE, seed = 18210)
fit <- with(data = imp, lm(bmi ~ age))

# manual pooling
summary(fit$analyses[[1]])
#> 
#> Call:
#> lm(formula = bmi ~ age)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -6.1587 -3.0674  0.9413  2.3870  8.7413 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  28.1043     1.8853   14.91 2.61e-13 ***
#> age          -1.5457     0.9723   -1.59    0.126    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 3.957 on 23 degrees of freedom
#> Multiple R-squared:  0.099,	Adjusted R-squared:  0.05983 
#> F-statistic: 2.527 on 1 and 23 DF,  p-value: 0.1255
#> 
summary(fit$analyses[[2]])
#> 
#> Call:
#> lm(formula = bmi ~ age)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -7.3611 -3.6333  0.9389  2.3389  7.5389 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   29.189      2.019  14.460 4.92e-13 ***
#> age           -1.428      1.041  -1.371    0.183    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 4.236 on 23 degrees of freedom
#> Multiple R-squared:  0.0756,	Adjusted R-squared:  0.03541 
#> F-statistic: 1.881 on 1 and 23 DF,  p-value: 0.1835
#> 
pool.scalar(Q = c(-1.5457, -1.428), U = c(0.9723^2, 1.041^2), n = 25, k = 2)
#> $m
#> [1] 2
#> 
#> $qhat
#> [1] -1.5457 -1.4280
#> 
#> $u
#> [1] 0.9453673 1.0836810
#> 
#> $qbar
#> [1] -1.48685
#> 
#> $ubar
#> [1] 1.014524
#> 
#> $b
#> [1] 0.006926645
#> 
#> $t
#> [1] 1.024914
#> 
#> $df
#> [1] 20.97025
#> 
#> $r
#> [1] 0.01024122
#> 
#> $fmi
#> [1] 0.09272831
#> 

# automatic pooling using broom
pool(fit)
#> Class: mipo    m = 2 
#>          term m  estimate     ubar           b        t dfcom       df
#> 1 (Intercept) 2 28.646618 3.814682 0.588114658 4.696854    23 10.72144
#> 2         age 2 -1.486715 1.014543 0.006947187 1.024964    23 20.96937
#>         riv     lambda        fmi
#> 1 0.2312570 0.18782190 0.30620278
#> 2 0.0102714 0.01016697 0.09275848