ABSTRACT
The purpose of this article was to introduce the χ2 distribution and related contents， including χ2 distribution and non-central χ2 distribution. It focused on showing the definition of two χ2 distributions， the graph and the main properties of the probability density function. Among them， the two most important properties were： first， the limiting distribution of the χ2 distribution was the normal distribution； second， n-1s2σ2followed the χ2 distribution with n-1 degrees of freedom.In addition， it also explained the relationship between the χ2 distribution and the normal distribution， the relationship between χ2 test statistic and Z test statistic. Finally， it illustrated the computational approaches of the χ2 distribution based on the two SAS functions in SAS software.
ABSTRACT
The purpose of this article was to introduce the goodness of fit test and its SAS implementation. The main contents included the following four aspects： ① Pearson΄s goodness of fit test； ② deviance or likelihood ratio goodness of fit test； ③ Hosmer-Lemeshow goodness of fit test； ④ goodness of fit test for the sparse data. In the aforementioned “fourth aspect”， there were six specific test approaches， namely “information matrix test” “information matrix diagonal test” “Osius-Rojek test” “unweighted residual sum of squares test” “Spiegelhalter test” and “Stukel test”. The paper implemented the four types of the goodness of fit tests mentioned above with the help of the SAS software through an example， explained the output results， and made statistical and professional conclusions.
ABSTRACT
The purpose of this article was to introduce the rank sum test and its SAS implementation. The concrete contents included the following three aspects： ①the simple linear rank test of two-sample data， ②the one-way ANOVA rank sum test of the multi-sample data， ③the "scoring methods" in the aforementioned two situations. In the third aspect above， there were 10 concrete scoring approaches. Based on an example and with the aid of the SAS software， the paper implemented the first-type rank sum test previously， explained the output results， and made statistical and professional conclusions.
ABSTRACT
The purpose of the paper was to introduce the three special tests of the survival data and the SAS implementation. Specifically， it was the multiple comparisons， the trend test and the covariate test of the survival data. The multiple comparisons involved two situations： "the pairwise comparison" and "the comparison with control group". In the trend test， it involved two algorithms： "the log-rank test" and "the Wilcoxon test". In the covariate test， it involved "the single covariate test method" and "the multi-covariate test method of adding one covariate step by step". With the help of the SAS software and based on an example， this article implemented the three special tests mentioned above， explained the output results， and made statistical and professional conclusions.
ABSTRACT
The purpose of this article was to introduce the likelihood ratio test， six nonparametric tests， and the SAS implementation of the survival data. Based on the assumption that the survival data had the exponential distribution， the likelihood ratio test method was derived， the main difference between six nonparametric test methods was that they had different weight functions. Under the conditions of non-stratification and stratification， the seven survival data hypothesis testing methods mentioned above could be used， and their common point was that their test statistics all followed the χ2 distribution. Through two examples and by means of the SAS software， the article realized the various hypothesis tests for two or more groups of survival data， outputed and explained SAS calculation results， and made statistical and professional conclusions.
ABSTRACT
The purpose of this article was to introduce the Wald΄s test and the SAS implementation. The specific contents involved the following nine aspects， namely the general Wald΄s test，the robust Wald΄s test，the constrained Wald΄s χ2 test，the generalized Wald΄s test，the generalized Wald΄s log-linear test，the Wald΄s F test，the Wald΄s adjusted F test，the Wald΄s log-linear F test and the adjusted Wald΄s log-linear F test. The paper used two examples and the SAS software to achieve most of the Wald΄s tests mentioned above.
ABSTRACT
The purpose of this article was to introduce the score test and the SAS implementation. Concretely， there were five common forms of the score test statistics， namely the general score test statistic，the robust score test statistic，the residual score test statistic，the generalized score test statistic，and the Lagrange multiplier test statistic. The score test was mainly used in the following three situations， namely， testing whether part or all of the regression coefficients were 0， testing whether a single effect was not in the regression model，and testing whether the parallel line hypothesis was established. The paper used two examples and the SAS software to realize the various score tests mentioned above， and focused on explaining the output results related to the score tests.
ABSTRACT
The purpose of this article was to introduce the likelihood ratio test and the SAS implementation. Specifically， three definitions of the likelihood ratio test statistics and six more commonly used likelihood ratio test statistics were introduced. The three definitions were called based on the spatial size of the parameter vector to construct the likelihood ratio test statistic， based on the two nested statistical models to construct the likelihood ratio test statistic and based on the full model and partial model to construct the likelihood ratio test statistic； the six commonly used likelihood ratio test statistics were the general likelihood ratio χ2 test statistic，the adjusted likelihood ratio χ2 test statistic，the profile likelihood ratio χ2 test statistic，the quasi-likelihood ratio χ2 test statistic，the pseudo-likelihood ratio χ2 test statistic，and the Rao-Scott likelihood ratio χ2 test statistic. In the paper， through two examples， and with the help of the SAS software， the likelihood ratio χ2 tests were realized， the SAS calculation results were output and explained， and statistical and professional conclusions were made.
ABSTRACT
The purpose of this article was to introduce the log-rank test and the SAS implementation. There were many different expression forms for the test statistics of the log-rank tests. Among them， there were two most common expression forms： the first was similar to "Pearson΄s goodness-of-fit χ2 test statistic"； the second was similar to "the test statistic of the odds ratio of high-dimensional table data， that was， the Breslow-Day΄s χ2 test statistic". The log-rank test statistic had two distribution types， one was the χ2 distribution， the other was the standard normal distribution. In the process of constructing log-rank test statistics， there were four contents that needed to be paid attention to： ① the sequential multiple four-fold tables formed by stratification or partition after sorting according to the individual "survival time"； ② it was necessary to distinguish whether the "survival time of each individual" was the complete data or the censored data； ③ only the calculation of the "theoretical or expected frequency" on a specific grid ［for example， （1，1） grid］ in each four-fold table data； ④ the method of calculating the theoretical frequency was different from the one in the independent test of the data of the four-fold table. Based on two examples with the different data structures， the paper realized the log-rank tests with the help of the SAS software.