From 8063da50b23b7b9896ae29099d9db171ddf5ca45 Mon Sep 17 00:00:00 2001 From: Thomas Debray <117118104+tdebray123@users.noreply.github.com> Date: Tue, 8 Oct 2024 16:05:36 +0200 Subject: [PATCH] clean up code --- _freeze/chapter_11/execute-results/html.json | 4 +- _freeze/chapter_11/figure-html/fig-hmr4-1.png | Bin 33006 -> 26402 bytes _freeze/chapter_11/figure-html/fig-hmr5-1.png | Bin 0 -> 67755 bytes chapter_11.qmd | 90 +++++++++++---- chapter_11_files/figure-html/fig-hmr4-1.png | Bin 33006 -> 26402 bytes chapter_11_files/figure-html/fig-hmr5-1.png | Bin 0 -> 67755 bytes docs/chapter_11.html | 103 ++++++++++-------- .../figure-html/fig-hmr4-1.png | Bin 33006 -> 26402 bytes .../figure-html/fig-hmr5-1.png | Bin 0 -> 67755 bytes docs/search.json | 4 +- 10 files changed, 134 insertions(+), 67 deletions(-) create mode 100644 _freeze/chapter_11/figure-html/fig-hmr5-1.png create mode 100644 chapter_11_files/figure-html/fig-hmr5-1.png create mode 100644 docs/chapter_11_files/figure-html/fig-hmr5-1.png diff --git a/_freeze/chapter_11/execute-results/html.json b/_freeze/chapter_11/execute-results/html.json index 04fb36b..30102fd 100644 --- a/_freeze/chapter_11/execute-results/html.json +++ b/_freeze/chapter_11/execute-results/html.json @@ -1,8 +1,8 @@ { - "hash": "2bb698836ad87a92f4ef8dfe3aada86f", + "hash": "5620e0a1aa55df1d350f2fd2eb875fcc", "result": { "engine": "knitr", - "markdown": "---\ntitle: \"Individual Participant Data Meta-analysis of clinical trials and real-world data\"\nauthors: \n - name: Pablo Verde\n affiliations:\n - ref: umcdusseldorf\n - name: Thomas Debray\n orcid: 0000-0002-1790-2719\n affiliations:\n - ref: smartdas\naffiliations:\n - id: smartdas\n name: Smart Data Analysis and Statistics B.V.\n city: Utrecht\n - id: umcdusseldorf\n name: Universitätsklinikum Düsseldorf\n city: Düsseldorf\nformat:\n html:\n toc: true\n number-sections: true\nexecute:\n cache: true\nbibliography: 'https://api.citedrive.com/bib/0d25b38b-db8f-43c4-b934-f4e2f3bd655a/references.bib?x=eyJpZCI6ICIwZDI1YjM4Yi1kYjhmLTQzYzQtYjkzNC1mNGUyZjNiZDY1NWEiLCAidXNlciI6ICIyNTA2IiwgInNpZ25hdHVyZSI6ICI0MGFkYjZhMzYyYWE5Y2U0MjQ2NWE2ZTQzNjlhMWY3NTk5MzhhNzUxZDNjYWIxNDlmYjM4NDgwOTYzMzY5YzFlIn0=/bibliography.bib'\n---\n\n\n\n\n## Introduction\n\n\n## Hierarchical Meta-Regression\nWe illustrate the implementation of hierarchical meta-regression using an example that involves the following data sources:\n\n* Aggregate data from 35 randomized trials investigating the efficacy of adjunctive treatments in managing diabetic foot problems compared with routine care\n* Individual participant data from a prospective cohort study investigating patient and limb survival in patients with diabetic foot ulcers \n\n### Aggregate data\nWe first retrieve the randomized evidence and summarize the treatment effect estimates using a random effects meta-analysis:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nlibrary(dplyr)\nlibrary(jarbes)\nlibrary(metafor)\n\ndata(\"healing\")\n\naddat <- escalc(measure=\"OR\", ai=y_t, bi=n_t-y_t, ci=y_c, di=n_c-y_c, data=healing)\n\nresults.ADJ <- metagen(TE = yi, seTE = sqrt(vi),\n studlab = Study, data = addat,\n sm = \"OR\", \n prediction = TRUE)\n```\n:::\n\n\nThe corresponding forest plot is depicted below. The endpoint is healing without amputations within a period less than or equal to 1 year.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](chapter_11_files/figure-html/unnamed-chunk-3-1.png){width=960}\n:::\n:::\n\n\nThe random effects meta-analysis yielded a pooled odds ratio of 1.89. However, substantial between-study heterogeneity was found, with $\\tau$ = 0.45.\n\n### Individual participant data\nSubsequently, we retrieve the individual participant data:\n\n\n::: {.cell}\n\n```{.r .cell-code}\ndata(\"healingipd\")\nIPD <- healingipd %>% dplyr::select(healing.without.amp, PAD, neuropathy,\n first.ever.lesion, no.continuous.care, \n male, diab.typ2, insulin, HOCHD, \n HOS, CRF, dialysis, DNOAP, smoking.ever, \n diabdur, wagner.class)\n```\n:::\n\n\nBriefly, these IPD were obtained from a prospective cohort study enrolling consecutive patients with diabetic foot ulcers (DFUs) and without previous major amputation in a single diabetes center between June 1998 and December 1999 [@morbach_long-term_2012]. The baseline characteristics of the study population are summarized below:\n\n\n::: {.cell}\n::: {.cell-output-display}\n\n```{=html}\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Healing without amputation
(N=165)
No healing without amputation
(N=95)
Overall
(N=260)
Age (years)
Mean (SD)69.1 (10.9)68.5 (11.0)68.9 (10.9)
Median [Min, Max]70.0 [25.0, 90.0]69.0 [36.0, 91.0]70.0 [25.0, 91.0]
Diabetes duration (years)
Mean (SD)15.9 (10.3)15.9 (11.2)15.9 (10.6)
Median [Min, Max]14.0 [1.00, 53.0]14.0 [0, 50.0]14.0 [0, 53.0]
Sex
Female71 (43.0%)35 (36.8%)106 (40.8%)
Male94 (57.0%)60 (63.2%)154 (59.2%)
Ever smoker
Yes97 (58.8%)57 (60.0%)154 (59.2%)
No68 (41.2%)38 (40.0%)106 (40.8%)
Diabetes type 2
Yes150 (90.9%)79 (83.2%)229 (88.1%)
No15 (9.1%)16 (16.8%)31 (11.9%)
Peripheral arterial disease
Yes82 (49.7%)66 (69.5%)148 (56.9%)
No83 (50.3%)29 (30.5%)112 (43.1%)
Neuropathy
Yes144 (87.3%)80 (84.2%)224 (86.2%)
No21 (12.7%)15 (15.8%)36 (13.8%)
First ever lesion
Yes70 (42.4%)44 (46.3%)114 (43.8%)
No95 (57.6%)51 (53.7%)146 (56.2%)
No continuous care
Yes115 (69.7%)62 (65.3%)177 (68.1%)
No50 (30.3%)33 (34.7%)83 (31.9%)
Insulin dependent
Yes109 (66.1%)65 (68.4%)174 (66.9%)
No56 (33.9%)30 (31.6%)86 (33.1%)
History of coronary events (CHD)
Yes31 (18.8%)21 (22.1%)52 (20.0%)
No134 (81.2%)74 (77.9%)208 (80.0%)
History of stroke
Yes36 (21.8%)19 (20.0%)55 (21.2%)
No129 (78.2%)76 (80.0%)205 (78.8%)
Charcot foot syndrome
Yes28 (17.0%)24 (25.3%)52 (20.0%)
No137 (83.0%)71 (74.7%)208 (80.0%)
Dialysis
Yes3 (1.8%)6 (6.3%)9 (3.5%)
No162 (98.2%)89 (93.7%)251 (96.5%)
Diabetic Neuropathic Osteoarthropathy (DNOAP)
Yes19 (11.5%)10 (10.5%)29 (11.2%)
No146 (88.5%)85 (89.5%)231 (88.8%)
Wagner score
1-2115 (69.7%)27 (28.4%)142 (54.6%)
3-4-550 (30.3%)68 (71.6%)118 (45.4%)
\n
\n```\n\n:::\n:::\n\n\nAs depicted above, IPD are available from 260 patients. Some of these patients have similar characteristics to those enrolled in the randomized trials. However, other patients have comorbidities, where one or more risk factors prevent them to participate in the RCTs due to ethical reasons. For example,\n118 patients have severe ulcer lesions (Wagner score 3 to 5), and 77 patients suffer from severe ulcer lesions and peripheral arterial disease (PAD). The question is: Can we generalize the benefit of adjuvant therapies observed in the RCTs to the subgroups of patients encountered in clinical practice?\n\n### Hierarchical metaregression\nWe first investigate the event rate of patients receiving routine care:\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](chapter_11_files/figure-html/unnamed-chunk-6-1.png){width=960}\n:::\n:::\n\n\nThe forest plot above indicates that the baseline risk in the observational study from Morbach et al. is much higher than most trials.\n\n\nWe fitted an HMR model to the available RWD and published AD: \n\n\n::: {.cell}\n\n```{.r .cell-code}\nset.seed(2022)\n\nAD <- healing %>% dplyr::select(yc = y_c, nc = n_c, \n yt = y_t, nt = n_t, Study = Study)\n\nmx2 <- hmr(data = AD, # Published aggregate data\n two.by.two = FALSE, # \n dataIPD = IPD, # Data frame of the IPD \n re = \"sm\", # Random effects model: \"sm\" scale mixtures \n link = \"logit\", # Link function of the random effects\n sd.mu.1 = 1, # Scale parameter for the prior of mu.1\n sd.mu.2 = 1, # Scale parameter for the prior of mu.2 \n sd.mu.phi = 1, # Scale parameter for the prior of mu.phi \n sigma.1.upper = 5, # Upper bound of the prior of sigma.1 \n sigma.2.upper = 5, # Upper bound of the prior of sigma.2\n sigma.beta.upper = 5, # Upper bound of the prior of sigma.beta\n sd.Fisher.rho = 1.25, # Scale parameter for the prior of rho\n df.estimate = TRUE, # If TRUE the degrees of freedom are estimated\n df.lower = 3, # Lower bound of the df's prior\n df.upper = 10, # Upper bound of the df's prior\n nr.chains = 2, # Number of MCMC chains\n nr.iterations = 10000, # Total number of iterations\n nr.adapt = 1000, # Number of iteration for burnin \n nr.thin = 1) # Thinning rate\n```\n:::\n\n\nWe start our analysis by visualizing the conflict of evidence between the different types of data and study types. The figure below depicts the posterior distribution of $\\mu_{\\phi}$, which is the mean bias of the IPD-NRS compared to the AD-RCTs control groups. With only one IPD-NRS, this parameter is partially\nidentifiable from the data. However, we can expect to learn about this bias parameter \nin a full Bayesian model.\n\n\n\n\n\n::: {.cell}\n\n```{.r .cell-code}\n# Pablo's calculations ...\nmu.phi <- mx2$BUGSoutput$sims.list$mu.phi\nmean(mu.phi)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.7868735\n```\n\n\n:::\n\n```{.r .cell-code}\nsd(mu.phi)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 1.091779\n```\n\n\n:::\n\n```{.r .cell-code}\n# Bias parameters: \n# mean sd 2.5% 25% 50% 75% 97.5% Rhat n.eff\n# mu.phi 0.79 1.09 -1.35 0.09 0.78 1.48 2.96 1 5000\n\n# Pr(mu.phi>0|Data) ...\n\nPr.mu.phi = sum(mu.phi > 0)/length(mu.phi)\nPr.mu.phi\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.7733889\n```\n\n\n:::\n:::\n\n\nThe posterior distribution of $\\mu_{\\phi}$ has a mean of 0.79 and a 95% posterior interval of [-1.35, 2.96]. The posterior probability that $\\mu_{\\phi}$ is greater than zero is 77%, which indicates \nthat in average the IPD-NRS of this cohort present a better prognoses that the AD-RCTs control groups.\nThat means that taking the IPD-NRS results at face value would be misleading if we aim to combine them with a meta-analysis of AD-RCTs.\n\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior sensitivity analysis of bias mean between the RCTs and the IPD-NRS.](chapter_11_files/figure-html/fig-hmr1-1.png){#fig-hmr1 width=1056}\n:::\n:::\n\n\n\n@fig-hmr2 presents the posterior distribution of the weights $w_{i}$ for each study included in the HMR. These posteriors are summarized using a forest plot, where posterior intervals substantially greater than one indicate outliers. One important aspect of the HMR is that those outliers are automatically down-weighted in the analysis.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior distribution of the study weights, illustrated by the median and 95% credible intervals. Studies with posterior weights greater than 1.5, marked in red, are flagged as potential outliers.](chapter_11_files/figure-html/fig-hmr2-1.png){#fig-hmr2 width=576}\n:::\n:::\n\n\n@fig-hmr3 displays the results of the submodel corresponding to the\nIPD-NRS that received only medical routine care. The posteriors of the\nregression coefficients $\\beta_k$ ($k=1,\\dots, 15$) are summarized in a forest plot. This submodel\ndetects risk factors that can reduce the chance of getting healed. We see that\nthe group of patients with a Wagner score greater than 2 have substantially less\nchance of getting healed compared to the group with lower scores. This can also\nbe observed in the group of patients with PAD.\n\nInterestingly, these subgroups of patients that have lower chances of getting\nhealed are underrepresented in the RCTs populations. Therefore, by combining\nIPD-NRS with AD-RCT we can learn new insights about these patients that cannot\nbe learned neither from AD nor from IPD alone.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior distribution of regression coefficients from the IPD-NRS analysis, illustrated by the mean and 95% credible intervals. The most relevant risk factors identified in this analysis were the Wagner classification (1-2 vs. 3-4-5) and the presence of peripheral arterial disease (PAD) (no vs. yes).](chapter_11_files/figure-html/fig-hmr3-1.png){#fig-hmr3 width=768}\n:::\n:::\n\n\nThe association between baseline healing risk without amputation within one year and the relative treatment effect is illustrated in @fig-hmr4. Results from the underlying HMR submodel are used to predict treatment effects across different patient subgroups, providing insights into how baseline risk impacts the effectiveness of the treatment. The posterior median and 95\\% credible intervals indicate that healthier patients (with a) are associated with a reduced treatment effect. In other words, healthier patients tend to derive less benefit from the adjunctive therapy compared to those with a higher baseline risk.\n\nThe model is centered at -0.565, corresponding to the posterior mean of $\\mu_1$, the RCTs' baseline risk. To the right of $\\mu_1$ we have the posterior mean of the IPD-NRS $\\mu_1 +\\mu_{\\phi}$, which has a posterior mean of 0.222. This shows an important bias captured by the introduction of $\\mu_{\\phi}$ in the model.\n\n\n::: {.cell}\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning in prop.test(x = healingplus$y_c[i], n = healingplus$n_c[i], correct =\nFALSE): Chi-squared approximation may be incorrect\n```\n\n\n:::\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning: Duplicated aesthetics after name standardisation: colour\n```\n\n\n:::\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.\nℹ Please use `linewidth` instead.\n```\n\n\n:::\n\n::: {.cell-output-display}\n![Summary results of generalizing relative treatment effects: The RCTs' results are displayed as a forest plot. The fitted hierarchical meta-regression model is summarized with solid lines representing the posterior median and 95% intervals. The predicted log odds ratio for the observational study is displayed in blue.](chapter_11_files/figure-html/fig-hmr4-1.png){#fig-hmr4 width=768}\n:::\n:::\n\n\n## Version info {.unnumbered}\nThis chapter was rendered using the following version of R and its packages:\n\n\n::: {.cell}\n::: {.cell-output .cell-output-stdout}\n\n```\nR version 4.4.1 (2024-06-14 ucrt)\nPlatform: x86_64-w64-mingw32/x64\nRunning under: Windows 10 x64 (build 19045)\n\nMatrix products: default\n\n\nlocale:\n[1] LC_COLLATE=English_United Kingdom.utf8 \n[2] LC_CTYPE=English_United Kingdom.utf8 \n[3] LC_MONETARY=English_United Kingdom.utf8\n[4] LC_NUMERIC=C \n[5] LC_TIME=English_United Kingdom.utf8 \n\ntime zone: Europe/Paris\ntzcode source: internal\n\nattached base packages:\n[1] stats graphics grDevices utils datasets methods base \n\nother attached packages:\n [1] meta_7.0-0 metadat_1.2-0 table1_1.4.3 tableone_0.13.2 \n [5] dplyr_1.1.4 jarbes_2.2.1 GGally_2.2.1 R2jags_0.8-5 \n [9] rjags_4-15 mcmcplots_0.4.3 coda_0.19-4.1 gridExtra_2.3 \n[13] ggplot2_3.5.1 kableExtra_1.4.0\n\nloaded via a namespace (and not attached):\n [1] tidyselect_1.2.1 viridisLite_0.4.2 farver_2.1.2 \n [4] fastmap_1.2.0 denstrip_1.5.4 CompQuadForm_1.4.3 \n [7] mathjaxr_1.6-0 promises_1.3.0 digest_0.6.36 \n[10] mime_0.12 lifecycle_1.0.4 survival_3.7-0 \n[13] magrittr_2.0.3 compiler_4.4.1 rlang_1.1.4 \n[16] tools_4.4.1 utf8_1.2.4 yaml_2.3.8 \n[19] knitr_1.48 labeling_0.4.3 htmlwidgets_1.6.4 \n[22] plyr_1.8.9 xml2_1.3.6 RColorBrewer_1.1-3 \n[25] abind_1.4-8 miniUI_0.1.1.1 withr_3.0.1 \n[28] purrr_1.0.2 numDeriv_2016.8-1.1 grid_4.4.1 \n[31] fansi_1.0.6 xtable_1.8-4 colorspace_2.1-0 \n[34] scales_1.3.0 MASS_7.3-61 cli_3.6.3 \n[37] survey_4.4-2 rmarkdown_2.28 metafor_4.6-0 \n[40] generics_0.1.3 rstudioapi_0.16.0 tzdb_0.4.0 \n[43] minqa_1.2.7 DBI_1.2.3 stringr_1.5.1 \n[46] splines_4.4.1 parallel_4.4.1 mitools_2.4 \n[49] vctrs_0.6.5 boot_1.3-31 Matrix_1.7-0 \n[52] jsonlite_1.8.8 hms_1.1.3 Formula_1.2-5 \n[55] systemfonts_1.1.0 tidyr_1.3.1 R2WinBUGS_2.1-22.1 \n[58] glue_1.7.0 nloptr_2.1.1 codetools_0.2-20 \n[61] ggstats_0.6.0 stringi_1.8.4 gtable_0.3.5 \n[64] later_1.3.2 sfsmisc_1.1-19 lme4_1.1-35.5 \n[67] munsell_0.5.1 tibble_3.2.1 pillar_1.9.0 \n[70] htmltools_0.5.8.1 ggExtra_0.10.1 R6_2.5.1 \n[73] evaluate_0.24.0 shiny_1.9.1 lattice_0.22-6 \n[76] readr_2.1.5 httpuv_1.6.15 Rcpp_1.0.13 \n[79] svglite_2.1.3 nlme_3.1-165 xfun_0.45 \n[82] pkgconfig_2.0.3 \n```\n\n\n:::\n:::\n\n\n## References {.unnumbered}\n\n", + "markdown": "---\ntitle: \"Individual Participant Data Meta-analysis of clinical trials and real-world data\"\nauthors: \n - name: Pablo Verde\n affiliations:\n - ref: umcdusseldorf\n - name: Thomas Debray\n orcid: 0000-0002-1790-2719\n affiliations:\n - ref: smartdas\naffiliations:\n - id: smartdas\n name: Smart Data Analysis and Statistics B.V.\n city: Utrecht\n - id: umcdusseldorf\n name: Universitätsklinikum Düsseldorf\n city: Düsseldorf\nformat:\n html:\n toc: true\n number-sections: true\nexecute:\n cache: true\nbibliography: 'https://api.citedrive.com/bib/0d25b38b-db8f-43c4-b934-f4e2f3bd655a/references.bib?x=eyJpZCI6ICIwZDI1YjM4Yi1kYjhmLTQzYzQtYjkzNC1mNGUyZjNiZDY1NWEiLCAidXNlciI6ICIyNTA2IiwgInNpZ25hdHVyZSI6ICI0MGFkYjZhMzYyYWE5Y2U0MjQ2NWE2ZTQzNjlhMWY3NTk5MzhhNzUxZDNjYWIxNDlmYjM4NDgwOTYzMzY5YzFlIn0=/bibliography.bib'\n---\n\n\n\n\n## Introduction\n\n\n## Hierarchical Meta-Regression\nWe illustrate the implementation of hierarchical meta-regression using an example that involves the following data sources:\n\n* Aggregate data from 35 randomized trials investigating the efficacy of adjunctive treatments in managing diabetic foot problems compared with routine care\n* Individual participant data from a prospective cohort study investigating patient and limb survival in patients with diabetic foot ulcers \n\n### Aggregate data\nWe first retrieve the randomized evidence and summarize the treatment effect estimates using a random effects meta-analysis:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nlibrary(dplyr)\nlibrary(jarbes)\nlibrary(metafor)\n\ndata(\"healing\")\n\naddat <- escalc(measure=\"OR\", ai=y_t, bi=n_t-y_t, ci=y_c, di=n_c-y_c, data=healing)\n\nresults.ADJ <- metagen(TE = yi, seTE = sqrt(vi),\n studlab = Study, data = addat,\n sm = \"OR\", \n prediction = TRUE)\n```\n:::\n\n\nThe corresponding forest plot is depicted below. The endpoint is healing without amputations within a period less than or equal to 1 year.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](chapter_11_files/figure-html/unnamed-chunk-3-1.png){width=960}\n:::\n:::\n\n\nThe random effects meta-analysis yielded a pooled odds ratio of 1.89. However, substantial between-study heterogeneity was found, with $\\tau$ = 0.45.\n\n### Individual participant data\nSubsequently, we retrieve the individual participant data:\n\n\n::: {.cell}\n\n```{.r .cell-code}\ndata(\"healingipd\")\nIPD <- healingipd %>% dplyr::select(healing.without.amp, PAD, neuropathy,\n first.ever.lesion, no.continuous.care, \n male, diab.typ2, insulin, HOCHD, \n HOS, CRF, dialysis, DNOAP, smoking.ever, \n diabdur, wagner.class)\n```\n:::\n\n\nBriefly, these IPD were obtained from a prospective cohort study enrolling consecutive patients with diabetic foot ulcers (DFUs) and without previous major amputation in a single diabetes center between June 1998 and December 1999 [@morbach_long-term_2012]. The baseline characteristics of the study population are summarized below:\n\n\n::: {.cell}\n::: {.cell-output-display}\n\n```{=html}\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Healing without amputation
(N=165)
No healing without amputation
(N=95)
Overall
(N=260)
Age (years)
Mean (SD)69.1 (10.9)68.5 (11.0)68.9 (10.9)
Median [Min, Max]70.0 [25.0, 90.0]69.0 [36.0, 91.0]70.0 [25.0, 91.0]
Diabetes duration (years)
Mean (SD)15.9 (10.3)15.9 (11.2)15.9 (10.6)
Median [Min, Max]14.0 [1.00, 53.0]14.0 [0, 50.0]14.0 [0, 53.0]
Sex
Female71 (43.0%)35 (36.8%)106 (40.8%)
Male94 (57.0%)60 (63.2%)154 (59.2%)
Ever smoker
Yes97 (58.8%)57 (60.0%)154 (59.2%)
No68 (41.2%)38 (40.0%)106 (40.8%)
Diabetes type 2
Yes150 (90.9%)79 (83.2%)229 (88.1%)
No15 (9.1%)16 (16.8%)31 (11.9%)
Peripheral arterial disease
Yes82 (49.7%)66 (69.5%)148 (56.9%)
No83 (50.3%)29 (30.5%)112 (43.1%)
Neuropathy
Yes144 (87.3%)80 (84.2%)224 (86.2%)
No21 (12.7%)15 (15.8%)36 (13.8%)
First ever lesion
Yes70 (42.4%)44 (46.3%)114 (43.8%)
No95 (57.6%)51 (53.7%)146 (56.2%)
No continuous care
Yes115 (69.7%)62 (65.3%)177 (68.1%)
No50 (30.3%)33 (34.7%)83 (31.9%)
Insulin dependent
Yes109 (66.1%)65 (68.4%)174 (66.9%)
No56 (33.9%)30 (31.6%)86 (33.1%)
History of coronary events (CHD)
Yes31 (18.8%)21 (22.1%)52 (20.0%)
No134 (81.2%)74 (77.9%)208 (80.0%)
History of stroke
Yes36 (21.8%)19 (20.0%)55 (21.2%)
No129 (78.2%)76 (80.0%)205 (78.8%)
Charcot foot syndrome
Yes28 (17.0%)24 (25.3%)52 (20.0%)
No137 (83.0%)71 (74.7%)208 (80.0%)
Dialysis
Yes3 (1.8%)6 (6.3%)9 (3.5%)
No162 (98.2%)89 (93.7%)251 (96.5%)
Diabetic Neuropathic Osteoarthropathy (DNOAP)
Yes19 (11.5%)10 (10.5%)29 (11.2%)
No146 (88.5%)85 (89.5%)231 (88.8%)
Wagner score
1-2115 (69.7%)27 (28.4%)142 (54.6%)
3-4-550 (30.3%)68 (71.6%)118 (45.4%)
\n
\n```\n\n:::\n:::\n\n\nAs depicted above, IPD are available from 260 patients. Some of these patients have similar characteristics to those enrolled in the randomized trials. However, other patients have comorbidities, where one or more risk factors prevent them to participate in the RCTs due to ethical reasons. For example,\n118 patients have severe ulcer lesions (Wagner score 3 to 5), and 77 patients suffer from severe ulcer lesions and peripheral arterial disease (PAD). The question is: Can we generalize the benefit of adjuvant therapies observed in the RCTs to the subgroups of patients encountered in clinical practice?\n\n### Hierarchical metaregression\nWe first investigate the event rate of patients receiving routine care:\n\n\n::: {.cell}\n::: {.cell-output-display}\n![](chapter_11_files/figure-html/unnamed-chunk-6-1.png){width=960}\n:::\n:::\n\n\nThe forest plot above indicates that the baseline risk in the observational study from Morbach et al. is much higher than most trials.\n\n\nWe fitted an HMR model to the available RWD and published AD: \n\n\n::: {.cell}\n\n```{.r .cell-code}\nset.seed(2022)\n\nAD <- healing %>% dplyr::select(yc = y_c, nc = n_c, \n yt = y_t, nt = n_t, Study = Study)\n\nmx2 <- hmr(data = AD, # Published aggregate data\n two.by.two = FALSE, # \n dataIPD = IPD, # Data frame of the IPD \n re = \"sm\", # Random effects model: \"sm\" scale mixtures \n link = \"logit\", # Link function of the random effects\n sd.mu.1 = 1, # Scale parameter for the prior of mu.1\n sd.mu.2 = 1, # Scale parameter for the prior of mu.2 \n sd.mu.phi = 1, # Scale parameter for the prior of mu.phi \n sigma.1.upper = 5, # Upper bound of the prior of sigma.1 \n sigma.2.upper = 5, # Upper bound of the prior of sigma.2\n sigma.beta.upper = 5, # Upper bound of the prior of sigma.beta\n sd.Fisher.rho = 1.25, # Scale parameter for the prior of rho\n df.estimate = TRUE, # If TRUE the degrees of freedom are estimated\n df.lower = 3, # Lower bound of the df's prior\n df.upper = 10, # Upper bound of the df's prior\n nr.chains = 2, # Number of MCMC chains\n nr.iterations = 10000, # Total number of iterations\n nr.adapt = 1000, # Number of iteration for burnin \n nr.thin = 1) # Thinning rate\n```\n:::\n\n\nWe start our analysis by visualizing the conflict of evidence between the different types of data and study types. The figure below depicts the posterior distribution of $\\mu_{\\phi}$, which is the mean bias of the IPD-NRS compared to the AD-RCTs control groups. With only one IPD-NRS, this parameter is partially\nidentifiable from the data. However, we can expect to learn about this bias parameter \nin a full Bayesian model.\n\n\n\n\n\n::: {.cell}\n\n```{.r .cell-code}\n# Pablo's calculations ...\nmu.phi <- mx2$BUGSoutput$sims.list$mu.phi\nmean(mu.phi)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.7868735\n```\n\n\n:::\n\n```{.r .cell-code}\nsd(mu.phi)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 1.091779\n```\n\n\n:::\n\n```{.r .cell-code}\n# Bias parameters: \n# mean sd 2.5% 25% 50% 75% 97.5% Rhat n.eff\n# mu.phi 0.79 1.09 -1.35 0.09 0.78 1.48 2.96 1 5000\n\n# Pr(mu.phi>0|Data) ...\n\nPr.mu.phi = sum(mu.phi > 0)/length(mu.phi)\nPr.mu.phi\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.7733889\n```\n\n\n:::\n:::\n\n\nThe posterior distribution of $\\mu_{\\phi}$ has a mean of 0.79 and a 95% posterior interval of [-1.35, 2.96]. The posterior probability that $\\mu_{\\phi}$ is greater than zero is 77%, which indicates \nthat in average the IPD-NRS of this cohort present a better prognoses that the AD-RCTs control groups.\nThat means that taking the IPD-NRS results at face value would be misleading if we aim to combine them with a meta-analysis of AD-RCTs.\n\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior sensitivity analysis of bias mean between the RCTs and the IPD-NRS.](chapter_11_files/figure-html/fig-hmr1-1.png){#fig-hmr1 width=1056}\n:::\n:::\n\n\n\n@fig-hmr2 presents the posterior distribution of the weights $w_{i}$ for each study included in the HMR. These posteriors are summarized using a forest plot, where posterior intervals substantially greater than one indicate outliers. One important aspect of the HMR is that those outliers are automatically down-weighted in the analysis.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior distribution of the study weights, illustrated by the median and 95% credible intervals. Studies with posterior weights greater than 1.5, marked in red, are flagged as potential outliers.](chapter_11_files/figure-html/fig-hmr2-1.png){#fig-hmr2 width=576}\n:::\n:::\n\n\n@fig-hmr3 displays the results of the submodel corresponding to the\nIPD-NRS that received only medical routine care. The posteriors of the\nregression coefficients $\\beta_k$ ($k=1,\\dots, 15$) are summarized in a forest plot. This submodel\ndetects risk factors that can reduce the chance of getting healed. We see that\nthe group of patients with a Wagner score greater than 2 have substantially less\nchance of getting healed compared to the group with lower scores. This can also\nbe observed in the group of patients with PAD.\n\nInterestingly, these subgroups of patients that have lower chances of getting\nhealed are underrepresented in the RCTs populations. Therefore, by combining\nIPD-NRS with AD-RCT we can learn new insights about these patients that cannot\nbe learned neither from AD nor from IPD alone.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior distribution of regression coefficients from the IPD-NRS analysis, illustrated by the mean and 95% credible intervals. The most relevant risk factors identified in this analysis were the Wagner classification (1-2 vs. 3-4-5) and the presence of peripheral arterial disease (PAD) (no vs. yes).](chapter_11_files/figure-html/fig-hmr3-1.png){#fig-hmr3 width=768}\n:::\n:::\n\n\nThe association between baseline healing risk without amputation within one year and the relative treatment effect is illustrated in @fig-hmr4. Results from the underlying HMR submodel are used to predict treatment effects across different patient subgroups, providing insights into how baseline risk impacts the effectiveness of the treatment. The posterior median and 95\\% credible intervals indicate that healthier patients (with a) are associated with a reduced treatment effect. In other words, healthier patients tend to derive less benefit from the adjunctive therapy compared to those with a higher baseline risk.\n\nThe model is centered at -0.565, corresponding to the posterior mean of $\\mu_1$, the RCTs' baseline risk. To the right of $\\mu_1$ we have the posterior mean of the IPD-NRS $\\mu_1 +\\mu_{\\phi}$, which has a posterior mean of 0.222. This shows an important bias captured by the introduction of $\\mu_{\\phi}$ in the model.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Summary results of generalizing relative treatment effects: The results of the RCTs are presented as a forest plot. The fitted hierarchical meta-regression model is depicted with solid lines representing the posterior median and 95% credible intervals.](chapter_11_files/figure-html/fig-hmr4-1.png){#fig-hmr4 width=768}\n:::\n:::\n\n\n@fig-hmr5 presents the posterior effectiveness contours of $(\\theta_{new,0}^l(B), \\delta_{new}^l(B))$ for the subgroups of patients not included in the RCTs and with low chances of getting healed. On the left panel we have the resulting contour for patients with PAD (i.e. $l=15$ and $\\beta_{15}$) and on the right panel for patients with Wagner score 3 and 4 (i.e. $l=1$ and $\\beta_1$).\n\nThe horizontal axis displays the uncertainty in the location of the baseline\nrisk $\\theta_{new,0}^l(B)$ of these subgroups. This uncertainty resulted from the\nposterior variability of $\\mu_1$, $\\mu_{\\phi}$, $\\beta_l$ and the amount of bias\ncorrection $B$. We can see that for both subgroups the posterior effectiveness\n$\\delta_{new}^l(B)$ is above the horizontal line of no effectiveness for the full range of $\\theta_{new,0}^l(B)$. If the clinical context is adequate, then these results indicate that these subgroup of patients may benefit from this new intervention.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Posterior contourns (50%, 75% and 95%) for the effectivenes for subgroups identified in the Hierarchical Meta-Regression analysis. Left panel: Subgroup of patients with PDA. Right panel: Subgroup of patients with Wagner score > 2.](chapter_11_files/figure-html/fig-hmr5-1.png){#fig-hmr5 width=768}\n:::\n:::\n\n\n\n\n## Version info {.unnumbered}\nThis chapter was rendered using the following version of R and its packages:\n\n\n::: {.cell}\n::: {.cell-output .cell-output-stdout}\n\n```\nR version 4.4.1 (2024-06-14 ucrt)\nPlatform: x86_64-w64-mingw32/x64\nRunning under: Windows 10 x64 (build 19045)\n\nMatrix products: default\n\n\nlocale:\n[1] LC_COLLATE=English_United Kingdom.utf8 \n[2] LC_CTYPE=English_United Kingdom.utf8 \n[3] LC_MONETARY=English_United Kingdom.utf8\n[4] LC_NUMERIC=C \n[5] LC_TIME=English_United Kingdom.utf8 \n\ntime zone: Europe/Paris\ntzcode source: internal\n\nattached base packages:\n[1] stats graphics grDevices utils datasets methods base \n\nother attached packages:\n [1] metafor_4.6-0 numDeriv_2016.8-1.1 Matrix_1.7-0 \n [4] baggr_0.7.8 Rcpp_1.0.13 meta_7.0-0 \n [7] metadat_1.2-0 table1_1.4.3 tableone_0.13.2 \n[10] dplyr_1.1.4 jarbes_2.2.1 GGally_2.2.1 \n[13] R2jags_0.8-5 rjags_4-15 mcmcplots_0.4.3 \n[16] coda_0.19-4.1 gridExtra_2.3 ggplot2_3.5.1 \n[19] kableExtra_1.4.0 \n\nloaded via a namespace (and not attached):\n [1] DBI_1.2.3 inline_0.3.19 testthat_3.2.1.1 \n [4] rlang_1.1.4 magrittr_2.0.3 matrixStats_1.4.1 \n [7] compiler_4.4.1 loo_2.8.0 systemfonts_1.1.0 \n [10] vctrs_0.6.5 stringr_1.5.1 pkgconfig_2.0.3 \n [13] crayon_1.5.3 fastmap_1.2.0 backports_1.5.0 \n [16] labeling_0.4.3 utf8_1.2.4 promises_1.3.0 \n [19] rmarkdown_2.28 tzdb_0.4.0 forestplot_3.1.3 \n [22] nloptr_2.1.1 R2WinBUGS_2.1-22.1 purrr_1.0.2 \n [25] xfun_0.45 jsonlite_1.8.8 later_1.3.2 \n [28] parallel_4.4.1 R6_2.5.1 StanHeaders_2.32.10\n [31] stringi_1.8.4 RColorBrewer_1.1-3 denstrip_1.5.4 \n [34] boot_1.3-31 brio_1.1.5 rstan_2.32.6 \n [37] knitr_1.48 readr_2.1.5 bayesplot_1.11.1 \n [40] httpuv_1.6.15 splines_4.4.1 tidyselect_1.2.1 \n [43] rstudioapi_0.16.0 abind_1.4-8 yaml_2.3.8 \n [46] codetools_0.2-20 miniUI_0.1.1.1 curl_5.2.1 \n [49] pkgbuild_1.4.4 lattice_0.22-6 tibble_3.2.1 \n [52] plyr_1.8.9 shiny_1.9.1 withr_3.0.1 \n [55] evaluate_0.24.0 gridGraphics_0.5-1 survival_3.7-0 \n [58] CompQuadForm_1.4.3 isoband_0.2.7 ggstats_0.6.0 \n [61] RcppParallel_5.1.9 survey_4.4-2 xml2_1.3.6 \n [64] pillar_1.9.0 stats4_4.4.1 checkmate_2.3.2 \n [67] generics_0.1.3 mathjaxr_1.6-0 hms_1.1.3 \n [70] rstantools_2.4.0 munsell_0.5.1 scales_1.3.0 \n [73] minqa_1.2.7 xtable_1.8-4 glue_1.7.0 \n [76] tools_4.4.1 lme4_1.1-35.5 fs_1.6.4 \n [79] grid_4.4.1 tidyr_1.3.1 mitools_2.4 \n [82] QuickJSR_1.3.1 colorspace_2.1-0 nlme_3.1-165 \n [85] sfsmisc_1.1-19 Formula_1.2-5 cli_3.6.3 \n [88] fansi_1.0.6 viridisLite_0.4.2 svglite_2.1.3 \n [91] V8_4.4.2 gtable_0.3.5 yulab.utils_0.1.7 \n [94] digest_0.6.36 ggrepel_0.9.6 ggplotify_0.1.2 \n [97] farver_2.1.2 htmlwidgets_1.6.4 htmltools_0.5.8.1 \n[100] lifecycle_1.0.4 mime_0.12 ggExtra_0.10.1 \n[103] MASS_7.3-61 \n```\n\n\n:::\n:::\n\n\n## References {.unnumbered}\n\n", "supporting": [], "filters": [ 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zU)PW270v>q2RISy7^UalKEjk+T?=c+zq}Gv7*=`uTJqWH76V&Rvf7?}p}eVV{~bN* zL-s`4d%Veh4)DWGAo4|fr=Fw5lB(5S9&Os9VnFL|&&bb?Pl!kMj^*Wv+kd)DAJG@g zKD)PPMw9TUVLi7J-!l%Ie#B*zd5MlwgLM6h%L^3S zXX6H@-=Rbmo-sx@hn=??1r{gDN*Sx0cZAgO5*WkTs3wRQiw4Mk#`xH*tULytOl3rT)~#?k9#e817%AbOD^BjJH%>k#w5P@zXmr-v6xhv=z&^T#@rEqp;-ilN&?W z0?|TiKT746PrFVms`{1hsfw?G12BvI8lVC=Z7a@qK$+oN{iJxa-i~m#A1rm;r~RS5 zW%rgNTDHf$!P{FW%IuRQl-OUi?r^{D z!n#-H`}#wF5vXp{L-$ByF<16)&#f^?o)~ z4m<}NoE#{pBM!x%vQr=YxbZ$_{MtaoE6qk8tdP*6o;sRv+9tYh#% merge(summary(addat) %>% select("Study", "yi", "ci.lb", "ci.ub"), by = "Study") - - mu.phi <- mx2$BUGSoutput$sims.list$mu.phi mu.1 <- mx2$BUGSoutput$sims.list$mu.1 -X.baseline <- c(mean(mu.1), mean(mu.1 + mu.phi)) -vlines <- data.frame(X.baseline = X.baseline, Study.Types = Study.Types) - + # Clean breaks for expit values expit_breaks <- c(0.01, 0.05, 0.10, 0.25, 0.5, 0.75, 0.9, 0.95) logit_breaks <- qlogis(expit_breaks) # logit transformation # Calculate the proportion of patients who healed without amputation -prop_healed <- nrow(healingipd %>% filter(healing.without.amp == 1)) / nrow(healingipd) +prop_healed <- as.numeric(expit(mx2$BUGSoutput$mean$mu.1 + mx2$BUGSoutput$mean$mu.phi)) #nrow(healingipd %>% filter(healing.without.amp == 1)) / nrow(healingipd) df_rwd <- data.frame(x = prop_healed, y = mean((a0.f + b0.f * log(prop_healed / (1 - prop_healed)) )), ylow = quantile((a0.f + b0.f * log(prop_healed / (1 - prop_healed)) ), 0.025), yhigh = quantile((a0.f + b0.f * log(prop_healed / (1 - prop_healed)) ), 0.975)) +df_rct <- data.frame(x = expit(summary_stats$statistics["mu.1","Mean"]), + y = mean((a0.f + b0.f * summary_stats$statistics["mu.1","Mean"] )), + ylow = quantile((a0.f + b0.f * log(prop_healed / (1 - prop_healed)) ), 0.025), + yhigh = quantile((a0.f + b0.f * log(prop_healed / (1 - prop_healed)) ), 0.975)) + +df_combined <- rbind( + transform(df_rwd, source = "RWD"), + transform(df_rct, source = "RCT") +) ggplot(dat.lines, aes(x = prx, y = OR)) + - geom_line(aes(x = prx, y = median.hat), colour = "black" ) + - geom_line(aes(x = prx, y = upper.hat), colour = "black", lty = 2) + - geom_line(aes(x = prx, y = lower.hat), colour = "black", lty = 2) + + geom_line(aes(x = prx, y = median.hat)) + + geom_line(aes(x = prx, y = upper.hat), lty = 2) + + geom_line(aes(x = prx, y = lower.hat), lty = 2) + scale_x_continuous(name = "Probability of Healing with Routine Medical Care") + scale_y_continuous(name = "Log Odds ratio for adjuvant therapy") + geom_pointrange(data = healingplus, aes(x = prop, @@ -478,12 +478,64 @@ ggplot(dat.lines, aes(x = prx, y = OR)) + ymin = ci.lb, ymax = pmin(ci.ub, 6)), lwd = 0.8, alpha = 0.25, position = position_jitter(width = 0.02)) + - geom_hline(yintercept = 0, colour = "black", size = 0.5, lty = 2, color= "grey") + - geom_pointrange(data = df_rwd, aes(x = x, y = y, ymin = ylow, ymax = yhigh), color = "blue", lwd = 1.5, alpha = 0.50) + # Add the point - theme_bw() + geom_hline(yintercept = 0, linewidth = 0.5, lty = 2, color= "grey") + + geom_pointrange(data = df_combined, aes(x = x, y = y, ymin = ylow, ymax = yhigh, color = source), + lwd = 1.5) + # Combine RWD and RCT estimates + scale_color_manual(name = "Data Source", + values = c("RWD" = "darkorange", "RCT" = "dodgerblue3"), # Assign colors + labels = c("RCT summary", "RWD summary")) + # Assign colors + theme_bw() + theme(legend.position = "top", legend.title = element_blank()) # Place the legend at the top ``` +@fig-hmr5 presents the posterior effectiveness contours of $(\theta_{new,0}^l(B), \delta_{new}^l(B))$ for the subgroups of patients not included in the RCTs and with low chances of getting healed. On the left panel we have the resulting contour for patients with PAD (i.e. $l=15$ and $\beta_{15}$) and on the right panel for patients with Wagner score 3 and 4 (i.e. $l=1$ and $\beta_1$). + +The horizontal axis displays the uncertainty in the location of the baseline +risk $\theta_{new,0}^l(B)$ of these subgroups. This uncertainty resulted from the +posterior variability of $\mu_1$, $\mu_{\phi}$, $\beta_l$ and the amount of bias +correction $B$. We can see that for both subgroups the posterior effectiveness +$\delta_{new}^l(B)$ is above the horizontal line of no effectiveness for the full range of $\theta_{new,0}^l(B)$. If the clinical context is adequate, then these results indicate that these subgroup of patients may benefit from this new intervention. + +```{r} +#| fig-cap: "Posterior contourns (50%, 75% and 95%) for the effectivenes for subgroups identified in the Hierarchical Meta-Regression analysis. Left panel: Subgroup of patients with PDA. Right panel: Subgroup of patients with Wagner score > 2." +#| label: fig-hmr5 +#| echo: FALSE +#| warning: FALSE +#| message: FALSE +#| results: hide +#| fig-width: 8 + +# PDA +p.PDA = effect(mx2, + title.plot = "Subgroup with PDA", + k = 1, # Regression coefficient + x.lim = c(-7, 2.5), + y.lim = c(-.5, 2.5), + y.lab = "Effectiveness", + x.lab = "Baseline risk (logit scale)", + kde2d.n= 30, S = 15000, color.line = "blue", + font.size.title = 8) + +# Wanger +p.Wagner = effect(mx2, + title.plot = "Subgroup with Wagner Score > 2", + k = 15, # Regression coefficient + x.lim = c(-7, 2.5), + y.lim = c(-.5, 2.5), + y.lab = "Effectiveness", + x.lab = "Baseline risk (logit scale)", + kde2d.n= 30, S = 15000, + color.line = "red", + display.probability = FALSE, + line.no.effect = 0, + font.size.title = 8) + +gridExtra::grid.arrange(p.PDA, p.Wagner, ncol = 2, nrow = 1) + +``` + + + ## Version info {.unnumbered} This chapter was rendered using the following version of R and its packages: diff 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zU)PW270v>q2RISy7^UalKEjk+T?=c+zq}Gv7*=`uTJqWH76V&Rvf7?}p}eVV{~bN* zL-s`4d%Veh4)DWGAo4|fr=Fw5lB(5S9&Os9VnFL|&&bb?Pl!kMj^*Wv+kd)DAJG@g zKD)PPMw9TUVLi7J-!l%Ie#B*zd5MlwgLM6h%L^3S zXX6H@-=Rbmo-sx@hn=??1r{gDN*Sx0cZAgO5*WkTs3wRQiw4Mk#`xH*tULytOl3rT)~#?k9#e817%AbOD^BjJH%>k#w5P@zXmr-v6xhv=z&^T#@rEqp;-ilN&?W z0?|TiKT746PrFVms`{1hsfw?G12BvI8lVC=Z7a@qK$+oN{iJxa-i~m#A1rm;r~RS5 zW%rgNTDHf$!P{FW%IuRQl-OUi?r^{D z!n#-H`}#wF5vXp{L-$ByF<16)&#f^?o)~ z4m<}NoE#{pBM!x%vQr=YxbZ$_{MtaoE6qk8tdP*6o;sRv+9tYh#Figure 7.4. Results from the underlying HMR submodel are used to predict treatment effects across different patient subgroups, providing insights into how baseline risk impacts the effectiveness of the treatment. The posterior median and 95% credible intervals indicate that healthier patients (with a) are associated with a reduced treatment effect. In other words, healthier patients tend to derive less benefit from the adjunctive therapy compared to those with a higher baseline risk.

The model is centered at -0.565, corresponding to the posterior mean of \(\mu_1\), the RCTs’ baseline risk. To the right of \(\mu_1\) we have the posterior mean of the IPD-NRS \(\mu_1 +\mu_{\phi}\), which has a posterior mean of 0.222. This shows an important bias captured by the introduction of \(\mu_{\phi}\) in the model.

-
-
Warning in prop.test(x = healingplus$y_c[i], n = healingplus$n_c[i], correct =
-FALSE): Chi-squared approximation may be incorrect
-
-
-
Warning: Duplicated aesthetics after name standardisation: colour
-
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-
Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
-ℹ Please use `linewidth` instead.
-
@@ -835,7 +824,23 @@

-Figure 7.4: Summary results of generalizing relative treatment effects: The RCTs’ results are displayed as a forest plot. The fitted hierarchical meta-regression model is summarized with solid lines representing the posterior median and 95% intervals. The predicted log odds ratio for the observational study is displayed in blue. +Figure 7.4: Summary results of generalizing relative treatment effects: The results of the RCTs are presented as a forest plot. The fitted hierarchical meta-regression model is depicted with solid lines representing the posterior median and 95% credible intervals. +
+ +
+
+ +

Figure 7.5 presents the posterior effectiveness contours of \((\theta_{new,0}^l(B), \delta_{new}^l(B))\) for the subgroups of patients not included in the RCTs and with low chances of getting healed. On the left panel we have the resulting contour for patients with PAD (i.e. \(l=15\) and \(\beta_{15}\)) and on the right panel for patients with Wagner score 3 and 4 (i.e. \(l=1\) and \(\beta_1\)).

+

The horizontal axis displays the uncertainty in the location of the baseline risk \(\theta_{new,0}^l(B)\) of these subgroups. This uncertainty resulted from the posterior variability of \(\mu_1\), \(\mu_{\phi}\), \(\beta_l\) and the amount of bias correction \(B\). We can see that for both subgroups the posterior effectiveness \(\delta_{new}^l(B)\) is above the horizontal line of no effectiveness for the full range of \(\theta_{new,0}^l(B)\). If the clinical context is adequate, then these results indicate that these subgroup of patients may benefit from this new intervention.

+
+
+
+
+
+ +
+
+Figure 7.5: Posterior contourns (50%, 75% and 95%) for the effectivenes for subgroups identified in the Hierarchical Meta-Regression analysis. Left panel: Subgroup of patients with PDA. Right panel: Subgroup of patients with Wagner score > 2.
@@ -869,40 +874,50 @@

Version info

[1] stats graphics grDevices utils datasets methods base other attached packages: - [1] meta_7.0-0 metadat_1.2-0 table1_1.4.3 tableone_0.13.2 - [5] dplyr_1.1.4 jarbes_2.2.1 GGally_2.2.1 R2jags_0.8-5 - [9] rjags_4-15 mcmcplots_0.4.3 coda_0.19-4.1 gridExtra_2.3 -[13] ggplot2_3.5.1 kableExtra_1.4.0 + [1] metafor_4.6-0 numDeriv_2016.8-1.1 Matrix_1.7-0 + [4] baggr_0.7.8 Rcpp_1.0.13 meta_7.0-0 + [7] metadat_1.2-0 table1_1.4.3 tableone_0.13.2 +[10] dplyr_1.1.4 jarbes_2.2.1 GGally_2.2.1 +[13] R2jags_0.8-5 rjags_4-15 mcmcplots_0.4.3 +[16] coda_0.19-4.1 gridExtra_2.3 ggplot2_3.5.1 +[19] kableExtra_1.4.0 loaded via a namespace (and not attached): - [1] tidyselect_1.2.1 viridisLite_0.4.2 farver_2.1.2 - [4] fastmap_1.2.0 denstrip_1.5.4 CompQuadForm_1.4.3 - [7] mathjaxr_1.6-0 promises_1.3.0 digest_0.6.36 -[10] mime_0.12 lifecycle_1.0.4 survival_3.7-0 -[13] magrittr_2.0.3 compiler_4.4.1 rlang_1.1.4 -[16] tools_4.4.1 utf8_1.2.4 yaml_2.3.8 -[19] knitr_1.48 labeling_0.4.3 htmlwidgets_1.6.4 -[22] plyr_1.8.9 xml2_1.3.6 RColorBrewer_1.1-3 -[25] abind_1.4-8 miniUI_0.1.1.1 withr_3.0.1 -[28] purrr_1.0.2 numDeriv_2016.8-1.1 grid_4.4.1 -[31] fansi_1.0.6 xtable_1.8-4 colorspace_2.1-0 -[34] scales_1.3.0 MASS_7.3-61 cli_3.6.3 -[37] survey_4.4-2 rmarkdown_2.28 metafor_4.6-0 -[40] generics_0.1.3 rstudioapi_0.16.0 tzdb_0.4.0 -[43] minqa_1.2.7 DBI_1.2.3 stringr_1.5.1 -[46] splines_4.4.1 parallel_4.4.1 mitools_2.4 -[49] vctrs_0.6.5 boot_1.3-31 Matrix_1.7-0 -[52] jsonlite_1.8.8 hms_1.1.3 Formula_1.2-5 -[55] systemfonts_1.1.0 tidyr_1.3.1 R2WinBUGS_2.1-22.1 -[58] glue_1.7.0 nloptr_2.1.1 codetools_0.2-20 -[61] ggstats_0.6.0 stringi_1.8.4 gtable_0.3.5 -[64] later_1.3.2 sfsmisc_1.1-19 lme4_1.1-35.5 -[67] munsell_0.5.1 tibble_3.2.1 pillar_1.9.0 -[70] htmltools_0.5.8.1 ggExtra_0.10.1 R6_2.5.1 -[73] evaluate_0.24.0 shiny_1.9.1 lattice_0.22-6 -[76] readr_2.1.5 httpuv_1.6.15 Rcpp_1.0.13 -[79] svglite_2.1.3 nlme_3.1-165 xfun_0.45 -[82] pkgconfig_2.0.3 + [1] DBI_1.2.3 inline_0.3.19 testthat_3.2.1.1 + [4] rlang_1.1.4 magrittr_2.0.3 matrixStats_1.4.1 + [7] compiler_4.4.1 loo_2.8.0 systemfonts_1.1.0 + [10] vctrs_0.6.5 stringr_1.5.1 pkgconfig_2.0.3 + [13] crayon_1.5.3 fastmap_1.2.0 backports_1.5.0 + [16] labeling_0.4.3 utf8_1.2.4 promises_1.3.0 + [19] rmarkdown_2.28 tzdb_0.4.0 forestplot_3.1.3 + [22] nloptr_2.1.1 R2WinBUGS_2.1-22.1 purrr_1.0.2 + [25] xfun_0.45 jsonlite_1.8.8 later_1.3.2 + [28] parallel_4.4.1 R6_2.5.1 StanHeaders_2.32.10 + [31] stringi_1.8.4 RColorBrewer_1.1-3 denstrip_1.5.4 + [34] boot_1.3-31 brio_1.1.5 rstan_2.32.6 + [37] knitr_1.48 readr_2.1.5 bayesplot_1.11.1 + [40] httpuv_1.6.15 splines_4.4.1 tidyselect_1.2.1 + [43] rstudioapi_0.16.0 abind_1.4-8 yaml_2.3.8 + [46] codetools_0.2-20 miniUI_0.1.1.1 curl_5.2.1 + [49] pkgbuild_1.4.4 lattice_0.22-6 tibble_3.2.1 + [52] plyr_1.8.9 shiny_1.9.1 withr_3.0.1 + [55] evaluate_0.24.0 gridGraphics_0.5-1 survival_3.7-0 + [58] CompQuadForm_1.4.3 isoband_0.2.7 ggstats_0.6.0 + [61] RcppParallel_5.1.9 survey_4.4-2 xml2_1.3.6 + [64] pillar_1.9.0 stats4_4.4.1 checkmate_2.3.2 + [67] generics_0.1.3 mathjaxr_1.6-0 hms_1.1.3 + [70] rstantools_2.4.0 munsell_0.5.1 scales_1.3.0 + [73] minqa_1.2.7 xtable_1.8-4 glue_1.7.0 + [76] tools_4.4.1 lme4_1.1-35.5 fs_1.6.4 + [79] grid_4.4.1 tidyr_1.3.1 mitools_2.4 + [82] QuickJSR_1.3.1 colorspace_2.1-0 nlme_3.1-165 + [85] sfsmisc_1.1-19 Formula_1.2-5 cli_3.6.3 + [88] fansi_1.0.6 viridisLite_0.4.2 svglite_2.1.3 + [91] V8_4.4.2 gtable_0.3.5 yulab.utils_0.1.7 + [94] digest_0.6.36 ggrepel_0.9.6 ggplotify_0.1.2 + [97] farver_2.1.2 htmlwidgets_1.6.4 htmltools_0.5.8.1 +[100] lifecycle_1.0.4 mime_0.12 ggExtra_0.10.1 +[103] MASS_7.3-61
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chapter was rendered using the following version of R and its packages:\n\n\nR version 4.4.1 (2024-06-14 ucrt)\nPlatform: x86_64-w64-mingw32/x64\nRunning under: Windows 10 x64 (build 19045)\n\nMatrix products: default\n\n\nlocale:\n[1] LC_COLLATE=English_United Kingdom.utf8 \n[2] LC_CTYPE=English_United Kingdom.utf8 \n[3] LC_MONETARY=English_United Kingdom.utf8\n[4] LC_NUMERIC=C \n[5] LC_TIME=English_United Kingdom.utf8 \n\ntime zone: Europe/Paris\ntzcode source: internal\n\nattached base packages:\n[1] stats graphics grDevices utils datasets methods base \n\nother attached packages:\n [1] metafor_4.6-0 numDeriv_2016.8-1.1 Matrix_1.7-0 \n [4] baggr_0.7.8 Rcpp_1.0.13 meta_7.0-0 \n [7] metadat_1.2-0 table1_1.4.3 tableone_0.13.2 \n[10] dplyr_1.1.4 jarbes_2.2.1 GGally_2.2.1 \n[13] R2jags_0.8-5 rjags_4-15 mcmcplots_0.4.3 \n[16] coda_0.19-4.1 gridExtra_2.3 ggplot2_3.5.1 \n[19] kableExtra_1.4.0 \n\nloaded via a namespace (and not attached):\n [1] DBI_1.2.3 inline_0.3.19 testthat_3.2.1.1 \n 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