# 统计代写|线性回归分析代写linear regression analysis代考|Missing at Random

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## 统计代写|线性回归分析代写linear regression analysis代考|Missing at Random

The most common solution to missing data problems is to delete either cases or variables so the resulting data set is complete. Many software packages delete partially missing cases by default, and fit regression models to the remaining, complete, cases. This is a reasonable approach as long as the fraction of cases deleted is small enough, and the cause of values being unobserved is unrelated to the relationships under study. This would include data lost through an accident like dropping a test tube, or making an illegible entry in a logbook. If the reason for not observing values depends on the values that would have been observed, then the analysis of data may require modeling the cause of the failure to observe values. For example, if values of a measurement are unrecorded if the value is less than the minimum detection limit of an instrument, then the value is missing because the value that should have been observed is too small. A simple expedient in this case that is sometimes helpful is to substitute a value less than or equal to the detection limit for the unobserved values. This expedient is not always entirely satisfactory because substituting, or imputing, a fixed value for the unobserved quantity can reduce the variation on the filled-in variable, and yield misleading inferences.

As a second example, suppose we have a clinical trial that enrolls subjects with a particular medical condition, assigns each subject a treatment, and then the subjects are followed for a period of time to observe their response, which may be time until a particular landmark occurs, such as improvement of the medical condition. Subjects who do not respond well to the treatment may drop out of the study early, while subjects who do well may be more likely to remain in the study. Since the probability of observing a value depends on the value that would have been observed, simply deleting subjects who drop out early can easily lead to incorrect inferences because the successful subjects will be overrepresented among those who complete the study.

In many clinical trials, the response variable is not observed because the study ends, not because of patient characteristics. In this case, we call the response times censored; so for each patient, we know either the time to the landmark or the time to censoring. This is a different type of missing data problem, and analysis needs to include both the uncensored and censored observations. Book-length treatments of censored survival data are given by Kalbfleisch and Prentice (1980) and Cox and Oakes (1984), among others.

## 统计代写|线性回归分析代写linear regression analysis代考|Alternatives

All the alternatives we briefly outline here require strong assumptions concerning the data that may be impossible to check in practice.

Suppose first that we combine the response and predictors into a single vector $Z$. We assume that the distribution of $Z$ is fully known, apart from unknown parameters. The simplest assumption is that $Z \sim N(\boldsymbol{\mu}, \Sigma)$. If we had reasonable estimates of $\boldsymbol{\mu}$ and $\Sigma$, then we could use (4.15) to estimate parameters for the regression of the response on the other terms. The EM algorithm (Dempster, Laird, and Rubin , 1977) is a computational method that is used to estimate the parameters of the known joint distribution based on data with missing values.

Alternatively, given a model for the data like multivariate normality, one could impute values for the missing data and then analyze the completed data as if it were fully observed. Multiple imputation carries this one step further by creating several imputed data sets that, according to the model used, are plausible, filled-in data sets, and then “average” the analyses of the filled-in data sets. Software for both imputation and the EM algorithm for maximum likelihood estimate is available in several standard statistical packages, including the “missing” package in S-plus and the “MI” procedure in SAS.

The third approach is more comprehensive, as it requires building a model for the process of interest and the missing data process simultaneously. Examples of this approach are given by Ibrahim, Lipsitz, and Horton (2001), and Tang, Little, and Raghunathan (2003).

The data described in Table 4.2 provides an example. Allison and Cicchetti (1976) presented data on sleep patterns of 62 mammal species along with several other possible predictors of sleep. The data were in turn compiled from several other sources, and not all values are measured for all species. For example, $P S$, the number of hours of paradoxical sleep, was measured for only 50 of the 62 species in the data set, and $G P$, the gestation period, was measured for only 58 of the species. If we are interested in the dependence of hours of sleep on the other predictors, then we have at least three possible responses, $P S, S W S$, and $T S$, all observed on only a subset of the species. To use case deletion and then standard methods to analyze the conditional distributions of interest, we need to assume that the chance of a value being missing does not depend on the value. For example, the four missing values of GP are missing because no one had (as of 1976) published this value for these species. Using the imputation or the maximum likelihood methods are alternatives for these data, but they require making assumptions like normality, which might be palatable for many of the variables if transformed to logarithmic scale. Some of the variables, like $P$ and $S E$ are categorical, so other assumptions beyond multivariate normality might be needed.

# 线性回归代写

## 有限元方法代写

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