From: Multi-cohort modeling strategies for scalable globally accessible prostate cancer risk tools
Type of logistic regression | Model form | Risk predictor |
---|---|---|
1.Pooled data, cohort ignored | logit P(yi = 1) = β0 + β ′ xi by logistic regression fit to i = 1, …, n total number of patients | \( \frac{\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)\right\}} \) |
2.Pooled data, cohort as random effect, median prediction | logit P(yic = 1) = β0 + β0c + β′xic, β0c~N(0, d), by generalized linear mixed-effects models (binomial with logistic link) fit to i = 1, …, nc patients in c = 1, …, C centers | \( \frac{\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)\right\}} \) |
3.Pooled data, cohort as random effect, mean prediction | logit P(yic = 1) = β0 + β0c + β′xic, β0c~N(0, d), by generalized linear mixed-effects models (binomial with logistic link) fit to i = 1, …, nc patients in c = 1, …, C centers | \( {\int}_{-\infty}^{\infty}\frac{\mathit{\exp}\left({\beta}_0+{\beta}_{0c}+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}_{0c}+{\beta}^{\prime }x\right)\right\}}f\left({\beta}_{0c}\right)d{\beta}_{0c}, \) with f(β0c) density of β0c~N(0, d) |
4.Meta-analysis, fixed effects by center | logit P(yi = 1) = β0 + β ′ xi, with \( {\beta}_k=\frac{\sum_{c=1}^C{w}_{kc}{\beta}_{kc}}{\sum_{c=1}^C{w}_{kc}},k=0,\dots, 9, \) βkc estimated by separate logistic regressions for each center c = 1, …, C, wkc = 1/ var (βkc), where var(βkc) is the within-center estimate of the variance of βkc. | \( \frac{\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)\right\}} \) |
5.Meta-analysis, random effects by center | logit P(yi = 1) = β0 + β ′ xi, with \( {\beta}_k=\frac{\sum_{c=1}^C{w}_{kc}{\beta}_{kc}}{\sum_{c=1}^C{w}_{kc}},k=0,\dots, 9, \) βkc estimated by separate logistic regressions for each center c = 1, …, C, wkc = 1/{var(βkc) + b}, where var(βkc) is the within-center estimate of the variance of βkc, and b the between-center estimate of variance based on a method-of-moments estimation. | \( \frac{\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)\right\}} \) |