Estimating compound expectation in a regression framework with the new cereg command
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1 Estimating compound expectation in a regression framework with the new cereg command 2015 Nordic and Baltic Stata Users Group meeting Celia García-Pareja Matteo Bottai Unit of Biostatistics, IMM, KI September 4th, 2015 Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
2 Outline 1 Introduction 2 Motivating example 3 Mean vs quantiles 4 Compound Expectation 5 Estimation of the CE 6 Data Example 7 Results 8 Conclusions Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
3 Introduction Introduction Statistics is about summarizing information contained in observed data. The most informative, representative and precise the summary is, the better. Typical summary measures to provide are, for example, the sample mean and the quantiles. Question Which summary measure is more suitable? How precise is the information it provides? Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
4 Motivating example Motivating example I Simulated data on 450 observations drawn from a chi square with 4 d.f.. sqreg c, q( ) reps(200) t Coef. Std. Err. t P> t [95% Conf. Interval] q10 _cons q25 _cons q50 _cons q75 _cons q90 _cons regress t t Coef. Std. Err. t P> t [95% Conf. Interval] _cons Remarks Quantiles provide information about the whole distribution whereas the mean just refers to the mass center. Estimation of low quantiles is more precise than that of high quantiles. Inference on the mean is better than in high quantiles but worse than in low quantiles. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
5 Motivating example Motivating example II Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
6 Mean vs quantiles Mean vs quantiles The mean... Summarizes the data in a single number and it is easy to interpret. Its inference is extremely sensitive to the presence of outliers. Is informative just in case there is little variability in the data. Quantiles... Provide a detailed picture of the underlying statistical distribution. Can be estimated with high precision in regions with high density of data. Provide information about single points of the distribution. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
7 Mean vs quantiles Mean vs quantiles The mean... Summarizes the data in a single number and it is easy to interpret. Its inference is extremely sensitive to the presence of outliers. Is informative just in case there is little variability in the data. Quantiles... Proposal Provide a detailed picture of the underlying statistical distribution. Can be estimated with high precision in regions with high density of data. Provide information about single points of the distribution. Combine both summary measures providing a bridge between mean and quantiles. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
8 Compound Expectation Compound Expectation I The conditional expectation of Y can be written in terms of its quantile function as µ(x) = E[Y x] = ydf Y (y x) = 1 0 Q Y (p x)dp. Given a set of specified proportions {0, λ 1, λ 2,..., λ K 1, 1}, we split µ(x) into components µ(x) = 1 0 Q Y (p x)dp = λ1 0 λ2 1 Q Y (p x)dp+ λ 1 Q Y (p x)dp+...+ Q Y (p x)dp. λ K 1 Each component µ k (x) = λ k λ k 1 Q Y (p x)dp measures the contribution of a fraction of the population to µ(x). Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
9 Compound Expectation Compound Expectation II We might also calculate the expectation of every k-th component µ k (x) = µ k(x) λ k λ k 1. µ(x) can be then expressed as a weighted average of these expectations µ(x) = Special interest application settings Distributions with large variability: K (λ k λ k 1 )µ k (x). k=1 The mean is not representative and the quantiles might be insufficient. Censored data: Lack of information in the upper tail: the components can be computed up to the last observed quantile. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
10 Estimation of the CE Estimation of the CE Suppose that the conditional quantile function can be estimated as a linear combination of a set of covariates of interest: Q Y (p x) = ˆβ 0p + ˆβ 1px ˆβ spx s = s ˆβ jp x j. j=0 Every component ˆµ k (x) can be expressed as ˆµ k (x) = λk λ k 1 QY (p x)dp = λk λ k 1 s ˆβ jp x j dp = j=0 ( s λk j=0 λ k 1 ˆβjp dp ) x j = s ˆB jk x j. j=0 Therefore, ˆB jk is the effect of the j-th covariate in the k-th component. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
11 Data Example Data Example 347 patients with metastatic renal carcinoma. Patients randomly assigned to either subcutaneous interferon-α (IFN) or oral medroxyprogesterone acetate (MPA). After the total follow-up time, 322 patients had died and the censoring rate was 7.2%. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
12 Results Results I: components vs the overall mean. cereg days trt, f(died) c( ) reps(50) Compound Expectation regression No. of subjects = 347 No. of failures = 322 Coef. Std. Err. z P> z [95% Conf. Interval] q1_25 trt _cons q25_50 trt _cons q50_60 trt _cons q60_70 trt _cons q70_85 trt _cons q85_99 trt _cons The overall life expectancy after treatment initiation for those who had MPA was days and for those who had IFN was days ( days of difference). Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
13 Results Results II: life expectancy in portions of the population. cereg days trt, f(died) c( ) reps(50) means Compound Expectation regression No. of subjects = 347 No. of faiures = 322 Coef. Std. Err. z P> z [95% Conf. Interval] M1_25 trt _cons M25_50 trt _cons M50_60 trt _cons M60_70 trt _cons M70_85 trt _cons M85_99 trt _cons Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
14 Conclusions Conclusions The compound expectation is a suitable summary measure in any scenario. It can be used in a regression framework and thus, it provides information about the effect of a set of covariates of interest. It represents a useful tool for groups comparison. In the presence of censoring, it can be computed up to the last observed quantile, avoiding extrapolation. Further work: Optimize the components width for every specific case, in order to achieve better inferences. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
15 Thank you for your attention. Celia García-Pareja, Matteo Bottai (KI) Compound Expectation with the cereg command September 4th, / 14
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