Estimating and pooling dose-response curves from published data
Biostatistics group Meeting, April 8th
Alessio Crippa
Department of Nutritional Epidemiology and Biostatistics
IMM - Institute of Environmental Medicine
Karolinska Institutet
Dose-response relationship
Plot of the response on the y-axis versus the dose on the x-axis
Response: physiological or biochemical response, risk, counts, ordered descriptive categories (e.g., severity of a lesion), or continuous measurements (e.g., blood pressure)
Dose: different levels of an exposure (drug concentration, cups of coffe/day, BMI, etc.
Goal: evaluate changes of the response across levels of the exposure
Modeling dose-response associations
Categorize quantitative exposure
Results are presented in a tabular form
| dose | cases | n | RR | SE |
|---|---|---|---|---|
| (ref) | \( a_0 \) | \( n_0 \) | 1 | - |
| \( d_1 \) | \( a_1 \) | \( n_1 \) | \( RR_1 \) | \( SE_1 \) |
| \( \vdots \) | \( \vdots \) | \( \vdots \) | \( \vdots \) | \( \vdots \) |
| \( d_k \) | \( a_k \) | \( n_k \) | \( RR_K \) | \( SE_K \) |
Evaluate changes of the response across levels of the exposure
Assessment of the most likely shape of the curve (i.e. linear, J-shaped, U-shaped, etc.)
Two stage procedure
- Study-specific dose-response models
\[ Y_{i,j} = f(x_{i,j}; \theta_j) + \epsilon_{i,j} \]
Several alternatives (splines, polynomials, Emax, Exponential, Linear, etc.)
\( \epsilon_{i,j} \) are not independent
\( cov(\epsilon_{i,j}) \) can be obtained from published data to efficiently estimate \( \theta_j \) and \( V_j = cov(\theta_j) \)
Pool study specific estimates:
\[ \boldsymbol{\hat{\beta_j}} \sim N_p(\boldsymbol{\beta}, \mathbf{V_j}+ \boldsymbol{\psi}) \]
Example: Alcohol intake and colorectal cancer
8 eligible prospective cohort studies participating in the Pooling
Project of Prospective Studies of Diet and Cancer.
(http://www.imm.ki.se/biostatistics/glst/)
require("dosresmeta")
require("rms")
web <- "http://alessiocrippa.altervista.org/data/"
alcohol_crc <- read.table(paste0(web, "ex_alcohol_crc.txt"))
head(alcohol_crc)
id type dose cases peryears logrr se
1 atm ir 0.000 28 22186 0.0000 NA
2 atm ir 1.829 38 43031 -0.4167 0.2511
3 atm ir 9.199 43 53089 -0.3956 0.2456
4 atm ir 22.857 32 45348 -0.4884 0.2634
5 atm ir 35.667 16 19791 -0.2790 0.3208
6 atm ir 58.426 27 19920 0.2023 0.2862
knots <- quantile(alcohol_crc$dose, c(.1, .5, .9))
spl <- dosresmeta(formula = logrr ~ rcs(dose, knots), type = type,
cases = cases, n = peryears, id = id, se = se, data = alcohol_crc)
summary(spl)
Call: dosresmeta(formula = logrr ~ rcs(dose, knots), id = id, type = type,
cases = cases, n = peryears, data = alcohol_crc, se = se)
Multivariate random-effects meta-analysis
Dimension: 2
Estimation method: REML
Variance-covariance matrix Psi: unstructured
Approximate covariance method: Greenland & Longnecker
Fixed-effects coefficients
Estimate Std. Error z Pr(>|z|) 95%ci.lb
rcs(dose, knots).dose -0.0014 0.0041 -0.3297 0.7416 -0.0095
rcs(dose, knots).dose' 0.0210 0.0103 2.0446 0.0409 0.0009
95%ci.ub
rcs(dose, knots).dose 0.0068
rcs(dose, knots).dose' 0.0411 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Chi2 model: X2 = 27.2269 (df = 2), p-value = 0.0000
Multivariate Cochran Q-test for heterogeneity:
Q = 14.5886 (df = 14), p-value = 0.4068
I-square statistic = 4.0%
8 studies, 16 observations, 2 fixed and 3 random-effects parameters
logLik AIC BIC
41.4301 -72.8603 -69.6650
dose RR 95%ci.lb 95%ci.ub
1 0 1.00 1.00 1.00
2 5 0.99 0.96 1.03
3 10 0.99 0.92 1.07
4 15 1.00 0.90 1.11
5 20 1.02 0.91 1.15
6 25 1.06 0.93 1.20
7 30 1.10 0.97 1.25
8 35 1.16 1.03 1.31
9 40 1.23 1.09 1.39
10 45 1.32 1.16 1.49
11 50 1.41 1.23 1.61
12 55 1.51 1.29 1.77
13 60 1.63 1.35 1.95
Dose-response meta-analysis of Antipsychotics
Randomized placebo-controlled studies of patients with schizophrenia or schizoaffective disorder
Outcome is the difference in PANSS (or BPRS) score
Standardized Mean Differences (SMD)
| dose | n | PANSS | SE |
|---|---|---|---|
| (ref) | \( n_0 \) | 0 | - |
| \( d_1 \) | \( n_1 \) | \( y_1 \) | \( SE_1 \) |
| \( \vdots \) | \( \vdots \) | \( \vdots \) | \( \vdots \) |
| \( d_k \) | \( n_k \) | \( y_K \) | \( SE_K \) |
Dose-response meta-analysis of Antipsychotics (2)
\[ SMD_{ij} = \frac{y_{ij} - y_{1j}}{s_{ij}} \; \; , i = 2, \dots, k_j - 1 ; j = 1, \dots, m \]
\[ Var \left( SMD_{ij} \right) = \frac{1}{n_{ij}} + \frac{1}{n_{1j}} + \frac{1}{2*n_j} \; \; , i = 1, \dots, k_j ; j = 1, \dots, m \]
\[ Cov \left( SMD_{ij}, SMD_{i'j} \right) = \frac{1}{n_{1j}} + \frac{ SMD_{ij} * SMD_{i'j}}{2*n_j} \; \; , i != i' ; j = 1, \dots, m \]
Two studies included
ID Author Dose N PBchange SDPBc smd vsmd
1 1 Cutler 2006 0 85 5.30 18.31 0.0000 0.00000
2 1 Cutler 2006 2 92 8.23 18.32 0.1600 0.02267
3 1 Cutler 2006 5 89 10.60 18.31 0.2894 0.02312
4 1 Cutler 2006 10 94 11.30 18.32 0.3276 0.02255
5 2 McEvoy 2007 0 107 2.33 26.10 0.0000 0.00000
6 2 McEvoy 2007 10 103 15.04 27.60 0.4803 0.01934
7 2 McEvoy 2007 15 103 11.73 26.20 0.3552 0.01921
8 2 McEvoy 2007 20 97 14.44 25.90 0.4576 0.01991
with the corresponding covariance matrices
$`1`
[,1] [,2] [,3]
[1,] 0.02267 0.01183 0.01184
[2,] 0.01183 0.02312 0.01190
[3,] 0.01184 0.01190 0.02255
$`2`
[,1] [,2] [,3]
[1,] 0.019336 0.009554 0.009614
[2,] 0.009554 0.019208 0.009544
[3,] 0.009614 0.009544 0.019910
Dose-response meta-analysis of Antipsychotics (3)
Dose was modeled with restricted cubic splines (3 knots)
A large grid of value for the knots location
Fit spline models with knots located at the 25, 50, and 75 quantiles of the dose distribution
\[ SMD_{ij} = \beta_1 x_{1ij} + \beta2 x_{2ij} + \epsilon{ij} \]Select the model which “best” fits the data (minimum AIC)
Provide the dose-response relationship
Target doses: ED50, near-maximal dose range (ED85; ED95)
\[ ED_p = min \{d \in (d_1, d_k): f(d) > f(d_1) + p\delta_{max} \} \]
knots <- with(dat, quantile(Dose, c(.25, .5, .75)))
spl <- dosresmeta(formula = smd ~ rcs(Dose, knots), id = ID, v = vsmd, data = dat,
covariance = "user", ucov = vi, method = "reml")
summary(spl)
Call: dosresmeta(formula = smd ~ rcs(Dose, knots), id = ID, v = vsmd,
data = dat, covariance = "user", method = "reml", ucov = vi)
Multivariate random-effects meta-analysis
Dimension: 2
Estimation method: REML
Variance-covariance matrix Psi: unstructured
Approximate covariance method: User defined
Fixed-effects coefficients
Estimate Std. Error z Pr(>|z|) 95%ci.lb
rcs(Dose, knots).Dose 0.0476 0.0087 5.4978 0.0000 0.0306
rcs(Dose, knots).Dose' -0.0329 0.0084 -3.9074 0.0001 -0.0495
95%ci.ub
rcs(Dose, knots).Dose 0.0646 ***
rcs(Dose, knots).Dose' -0.0164 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Chi2 model: X2 = 50.1051 (df = 2), p-value = 0.0000
Multivariate Cochran Q-test for heterogeneity:
Q = 2.1727 (df = 8), p-value = 0.9753
I-square statistic = 1.0%
5 studies, 10 observations, 2 fixed and 3 random-effects parameters
logLik AIC BIC
23.5446 -37.0892 -36.6920
dose pred ci.lb ci.ub
1 0 0.00 0.00 0.00
2 5 0.23 0.15 0.31
3 10 0.39 0.26 0.52
4 15 0.44 0.31 0.57
5 20 0.42 0.31 0.54
6 25 0.41 0.28 0.53
7 30 0.39 0.23 0.54
p ED Ep
1 0.20 1.832 0.08764
2 0.50 4.775 0.21910
3 0.85 9.189 0.37247
4 0.95 11.502 0.41629
p EDspl EDEemax ED2 EDpol
1 0.20 1.83 0.24 2.16 5.0
2 0.50 4.77 0.96 6.04 9.5
3 0.85 9.19 4.68 12.61 15.5
4 0.95 11.50 11.50 15.98 18.5
References
Greenland, Sander, and Matthew P. Longnecker. “Methods for trend estimation from summarized dose-response data, with applications to meta-analysis.” American journal of epidemiology 135.11 (1992): 1301-1309.
Orsini, Nicola, et al. “Meta-analysis for linear and nonlinear dose-response relations: examples, an evaluation of approximations, and software.” American journal of epidemiology 175.1 (2012): 66-73.
Davis, J. M., & Chen, N. (2004). Dose response and dose equivalence of antipsychotics. Journal of clinical psychopharmacology, 24(2), 192-208.
Bornkamp, B., Pinheiro, J., & Bretz, F. (2009). MCPMod: An R package for the design and analysis of dose-finding studies. Journal of Statistical Software, 29(7), 1-23.