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Table 1 Five methods for optimal prediction in the validation; \( logit\ x=\mathit{\log}\left\{\frac{x}{1-x}\right\} \), yi = 1 high-grade cancer, 0 otherwise and xi = vector of covariates for the i th individual for all individuals across all centers (n the total number of individuals), β0 a fixed intercept, β = (β1, …, βk, …, β9) a fixed vector of parameters of length 9 for the covariates log2PSA, age, DRE, African ancestry, family history and prior negative biopsy history, as well as the interactions log2PSA and DRE, age and DRE, age and African ancestry

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 + βxic, 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 + βxic, 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. \( \frac{\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)}{\left\{1+\mathit{\exp}\left({\beta}_0+{\beta}^{\prime }x\right)\right\}} \)