Code
data(rmb_datasets, package = "rmb")
rmb_datasets$study_design[rmb_datasets$object == "fit"]
#> [1] "Randomized Fracture Intervention Trial of alendronate in postmenopausal women."FIT data
This article fits a logistic regression model to the Fracture Intervention Trial (FIT) dataset, evaluating whether alendronate reduces new vertebral fracture incidence in postmenopausal women (RMB2e Chapter 6).
1 Introduction
Postmenopausal osteoporosis is characterized by rapid bone loss driven by estrogen deficiency, leading to increased fracture risk. The Fracture Intervention Trial (FIT) was a large randomized placebo-controlled trial of the bisphosphonate alendronate in postmenopausal women with low femoral neck BMD (Black et al. 1996). New vertebral fractures were identified from serial lateral spine radiographs and serve as the primary binary outcome in this analysis. A history of prior vertebral fracture and low baseline BMD are strong confounders and are included as adjusters (RMB2e Ch. 6).
Does alendronate treatment reduce the incidence of new vertebral fractures compared to placebo in postmenopausal women with low BMD?
1.1 Causal assumptions
Code
set.seed(42)
dag <- ggdag::dagify(
vfx ~ treat + age + fracbase + bmdbase,
treat ~ bmdbase,
labels = c(
vfx = "New vertebral fx",
treat = "Alendronate",
age = "Age",
fracbase = "Prior fx",
bmdbase = "Baseline BMD"
),
exposure = "treat",
outcome = "vfx"
)
ggdag::ggdag(dag, use_labels = "label", text = FALSE) +
ggdag::theme_dag_blank() +
ggplot2::labs(title = "FIT: Causal DAG")
2 Methods
2.1 Study sample
Code
data(fit, package = "rmb")
dat <- fit
dim(dat)
#> [1] 5324 9
summary(haven::zap_labels(dat[c("treat", "age", "frac_base", "bmd_base", "newvfx", "fitpy")]))
#> treat age frac_base bmd_base
#> Min. :0.0000 Min. :55.05 Min. :0.0000 Min. : 5.432
#> 1st Qu.:0.0000 1st Qu.:64.00 1st Qu.:0.0000 1st Qu.: 8.509
#> Median :0.0000 Median :68.33 Median :0.0000 Median : 9.278
#> Mean :0.4987 Mean :68.27 Mean :0.3116 Mean : 9.183
#> 3rd Qu.:1.0000 3rd Qu.:72.70 3rd Qu.:1.0000 3rd Qu.: 9.969
#> Max. :1.0000 Max. :81.87 Max. :1.0000 Max. :12.292
#> newvfx fitpy
#> Min. :0.00000 Min. :2.478
#> 1st Qu.:0.00000 1st Qu.:3.020
#> Median :0.00000 Median :4.038
#> Mean :0.05522 Mean :3.849
#> 3rd Qu.:0.00000 3rd Qu.:4.487
#> Max. :1.00000 Max. :4.8192.2 Statistical analysis
Logistic regression models the probability of new vertebral fracture on treatment assignment, baseline fracture history, baseline femoral neck BMD, and age (RMB2e Ch. 6 / Black 1996).
Code
formula_main <- newvfx ~ treat + age + frac_base + bmd_base
formula_main
#> newvfx ~ treat + age + frac_base + bmd_base3 Results
3.1 Descriptive statistics
Code
with(dat, table(treat, newvfx))
#> newvfx
#> treat 0 1
#> 0 2474 195
#> 1 2556 99
with(dat, prop.table(table(treat, newvfx), margin = 1))
#> newvfx
#> treat 0 1
#> 0 0.92693893 0.07306107
#> 1 0.96271186 0.03728814
with(dat, tapply(bmd_base, treat, summary))
#> $`0`
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 5.432 8.509 9.278 9.186 9.985 11.727
#>
#> $`1`
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 5.667 8.525 9.294 9.180 9.953 12.292Code
fx_prop <- with(dat, prop.table(table(treat, newvfx), margin = 1))[, "1"]
fx_df <- data.frame(
treat = c("Placebo", "Alendronate"),
proportion = as.numeric(fx_prop)
)
fx_df$treat <- factor(fx_df$treat, levels = c("Placebo", "Alendronate"))
ggplot2::ggplot(fx_df, ggplot2::aes(x = treat, y = proportion, fill = treat)) +
ggplot2::geom_col() +
ggplot2::scale_fill_manual(values = c("#1b9e77", "#d95f02")) +
ggplot2::labs(
title = "FIT: Crude vertebral fracture rate by treatment",
x = NULL,
y = "Proportion with new vertebral fx"
) +
ggplot2::theme_minimal() +
ggplot2::theme(legend.position = "none")
3.2 Model estimates
Code
fit_model <- stats::glm(formula_main, data = dat, family = stats::binomial())
summary(fit_model)
#>
#> Call:
#> stats::glm(formula = formula_main, family = stats::binomial(),
#> data = dat)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -4.54016 1.04820 -4.331 1.48e-05 ***
#> treat -0.74296 0.13023 -5.705 1.16e-08 ***
#> age 0.06572 0.01158 5.673 1.40e-08 ***
#> frac_base 1.23173 0.13294 9.266 < 2e-16 ***
#> bmd_base -0.35406 0.05890 -6.011 1.84e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 2274.5 on 5323 degrees of freedom
#> Residual deviance: 1993.8 on 5319 degrees of freedom
#> AIC: 2003.8
#>
#> Number of Fisher Scoring iterations: 63.3 Model diagnostics
Code
fit_unadj <- stats::glm(newvfx ~ treat, data = dat, family = stats::binomial())
stats::anova(fit_unadj, fit_model, test = "Chisq")
#> Analysis of Deviance Table
#>
#> Model 1: newvfx ~ treat
#> Model 2: newvfx ~ treat + age + frac_base + bmd_base
#> Resid. Df Resid. Dev Df Deviance Pr(>Chi)
#> 1 5322 2241.3
#> 2 5319 1993.8 3 247.54 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 13.4 Inference
Code
or <- exp(stats::coef(fit_model))
ci <- exp(stats::confint(fit_model))
data.frame(
term = names(or),
odds_ratio = unname(or),
conf_low = ci[, 1],
conf_high = ci[, 2],
p_value = summary(fit_model)$coefficients[, "Pr(>|z|)"]
)
#> term odds_ratio conf_low conf_high p_value
#> (Intercept) (Intercept) 0.01067166 0.001350607 0.08236414 1.481647e-05
#> treat treat 0.47570534 0.367351009 0.61239533 1.164759e-08
#> age age 1.06792402 1.044124872 1.09265568 1.404115e-08
#> frac_base frac_base 3.42716051 2.647038329 4.45971891 1.940830e-20
#> bmd_base bmd_base 0.70183113 0.625168747 0.78763335 1.839665e-094 Discussion
Alendronate treatment is associated with substantially reduced odds of new vertebral fracture compared to placebo after adjustment for baseline fracture history, BMD, and age, replicating the main finding of Black et al. (1996) and discussed in RMB2e Chapter 6. Prior vertebral fracture and lower baseline BMD are both strong independent predictors of incident fracture, confirming their roles as established risk factors. These results illustrate how logistic regression can estimate treatment effects adjusted for important prognostic baseline characteristics in a randomized trial.
5 Source
- Black DM et al. (1996). Randomised trial of effect of alendronate on risk of fracture in women with existing vertebral fractures. Lancet, 348(9041), 1535–1541.
- UCSF Regression Methods companion data: https://regression.ucsf.edu/sites/g/files/tkssra16191/files/wysiwyg/home/data/fit.dta
- Book: Vittinghoff E, Glidden DV, Shiboski SC, McCulloch CE (2012). Regression Methods in Biostatistics (2nd edition).