ABSTRACT
The purpose of the paper was to introduce the multiple comparison method among multiple means and the SAS implementation. The multiple comparison approaches could be subdivided into the pairwise comparisons， the comparisons of all treatment groups with a control group， the comparisons of the mean of each treatment group with the average of all groups， the approximate and simulation-based approach， the multi-stage testing and Bayesian method. Except for the Bayesian approach， the difference between other multiple comparison methods lied in the types of error that were controlled. Error types could be roughly divided into the following three categories， the comparisonwise error rate， the experimentwise error rate and the maximum experimentwise error rate. The multiple comparison methods constructed based on the control of different error rates were not all the same in the strength of inference to draw conclusions. This paper used the SAS software to analyze the examples and explained the output results.
ABSTRACT
The purpose of this paper was to introduce the prerequisites， basic ideas， calculation formulas and the SAS implementation of a single-factor multi-level design quantitative data univariate analysis of variance. The prerequisites included the independence， normality and homogeneity of variance. The core of the basic idea was the decomposition of the sum of squares of the total deviations for the mean. The test statistic F was constructed through the between-group mean square divided by the within-group （or called error） mean square. The result of analysis of variance was a general evaluation of the difference among all means of a factor with the whole levels. When it was found that the difference among all means of the factor was statistically significant， a specific approach needed to be adopted for the multiple comparisons about the multiple means of the factor. With the help of the SAS software， the paper performed the analysis of variances for two examples， and used three approaches to make the multiple comparisons among all means of a factor in one of the examples.
ABSTRACT
The purpose of the paper was to introduce the test for homoscedasticity and the SAS implementation. The test of homogeneity of variance could be divided into the following three categories， ①analysis of variance directly based on comparison of variances， ②analysis of variance based on mean comparison was adopted for the new data from the variable transformation of the original data， ③the method of the χ2 test was used to analyze the quantitative raw data which followed the normal distribution. In the first category， a test statistic that followed the F distribution was constructed directly based on the variance ratio of the two samples. In the second category， there were a variety of different variable transformation approaches for the original data， and the new data after the transformation， which was viewed as the univariate quantitative data collected from a single-factor with multi-level design， was analyzed by using one-way ANOVA. In the third category， the χ2 test statistic was constructed for quantitative data that followed the normal distribution. The paper was based on SAS software to test the homogeneity of variances of three examples， and explained the output results.
ABSTRACT
The purpose of this paper was to outline the analysis of variance. Analysis of variance was a very important branch of statistics with rich contents and wide applicability. This paper summarized the analysis of variance from the following four aspects. Firstly， the basic concepts related to the analysis of variance. Secondly， the mathematical fundamentals of the analysis of variance， the F distribution. Thirdly， the application of the analysis of variance in the difference tests. Fourthly， the basic idea of the analysis of variance based on the comparison of means. In the first aspect， the definition， nature， meaning and contents of variance were mainly introduced. In the second aspect， the definition and nature of the F distribution were mainly introduced. In the third aspect， the analysis of variance in three kinds of application， namely the mean comparisons， the variance comparisons and the linear regression model evaluation were mainly introduced. In the fourth aspect， the core contents of the analysis of variance was the decomposition of the sum of squares of the total deviation from the mean and the construction of the test statistic F.
ABSTRACT
The purpose of this paper was to introduce the calculation formulas and the SAS implementation of the analysis of variance of the univariate quantitative data with the Latin square design. The Latin square design could be divided into two categories： the general Latin square design and the Greek Latin square design. The former could be used for the experimental situation with one experimental factor and two block factors， the latter could be used for the experimental situation with two experimental factors and two block factors. In fact， Latin square designs could be further subdivided by whether or not the repeated experiments were performed and whether the block factor was a single individual type. Generally speaking， in addition to satisfying the requirements of "independence， normality and homogeneity of variance"， the interaction between all factors was required to be non-existent or negligible when performing an analysis of variance on the quantitative data with Latin square design. When the quantitative data did not meet the preconditions mentioned above， it was recommended to use a mixed-effects model to build the model and solve it， or to solve the estimated values of the parameters in the ANOVA model based on the generalized estimating equation method.
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The purpose of the paper was to introduce the calculation formulas and the SAS implementation of the analysis of variance for the quantitative data of the crossover design. In the calculation， three test statistics were involved， namely Ftreatment， Fstage and Findividual. They were three test statistics used to evaluate the statistical significance of the effect of the treatment factor， the stage factor， and the individual factor on the quantitative outcome variable， respectively. In general， it was assumed that there was no or negligible interaction among the three factors in a crossover design， so there was no need to evaluate whether the interaction term was statistically significant. With the help of SAS software， this paper conducted the univariate analysis of variance for the quantitative data of crossover designs for three examples of 2×2 crossover design， 3×3 crossover design and three-stage crossover design， and presented the calculation results and drew the statistical and professional conclusions.
ABSTRACT
The purpose of this paper was to introduce the model， calculation formulas and the SAS implementation of the analysis of variance for the quantitative data with balanced incomplete block design. In the calculation， two test statistics were involved， namely FA and FB. Among them， the subscript "A" represented the experimental factor， and the subscript "B" represented the block factor B （i.e.， the important non-experimental factor）. In general， it was assumed that there was no or negligible interaction between the two factors in a balanced incomplete block design， so there was no need to evaluate whether the interaction term was statistically significant. Therefore， it was not necessary to do repeated experiments under each combination of two factors. With the help of SAS software， this paper conducted the analysis of variance for the quantitative data with balanced incomplete block design on two examples， and presented the calculation results and made the statistical and professional conclusions.
ABSTRACT
The purpose of this paper was to introduce the model， calculation formulas and the SAS implementation of the univariate analysis of variance for the quantitative data with randomized complete block design. In the calculation， two test statistics were involved， namely FA and FB. Among them， the subscript "A" represented the experimental factor， and the subscript "B" represented the block factor （i.e.， the important non-experimental factor）. In general， it was assumed that there was no or negligible interaction between the two factors in a randomized block design， so there was no need to assess whether the interaction term was statistically significant. Therefore， it was not necessary to do repeated experiments under each combination of two factors. With the help of SAS software， this paper conducted the analysis of variance for the quantitative data with randomized complete block design for two instances without and with repeated experiments， gave the calculation results， and made the statistical and professional conclusions.
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The purpose of this paper was to introduce the nested design and its quantitative data analysis of variance and the SAS implementation. If one of the following two characteristics existed in a specific experimental study， a nested design could be considered to arrange the experiment. Firstly， there was a nested relationship between factors in natural attributes. Secondly， with professional knowledge as the basis， the impact of each factor on the quantitative observation results was divided into primary and secondary. The first feature mentioned above meant that the factors related to the subjects had the conditions for grouping and regrouping. The second feature mentioned above meant that the status of each factor was unequal. In the variance analysis of quantitative data， the calculation formulas of variable error mean square was required to use. Based on four examples and with the help of the SAS software， this paper implemented the univariate analysis of variance for the quantitative data of the nested design， and gave the detailed explanations for the output results of SAS software.
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The purpose of this paper was to introduce the fractional factorial design and its quantitative data analysis of variance and the SAS implementation. The fractional factorial designs were very similar to the factorial designs and the orthogonal designs， but they had some differences. The fractional factorial design required significantly fewer combinations of levels than the factorial design of the same size， and even saved sample size than the orthogonal design of the same size. In general， the precision of the results obtained by a fractional factorial design was lower than an orthogonal design and much lower than a factorial design. The fractional factorial design was suitable for the trial tests with many experimental factors， and its main purpose was to explore experimental factors that had a greater impact on the quantitative experimental results. When performing ANOVA and regression analysis on quantitative data with a fractional factorial design， it should be clear which factors or interactions had confounded effects.
ABSTRACT
The purpose of this paper was to introduce the factorial design and its quantitative data analysis of variance and the SAS implementation. Factorial design could not only present the main effect magnitude of all experimental factors， but also comprehensively reflected the size of each-order interaction effect among multiple factors. However， this design required a large sample size. This paper introduced the calculation formulas of the analysis of variance for quantitative data with two-factor factorial design， and realized the analysis of variance for quantitative data with two-factor and three-factor factorial design through two examples with the help of SAS software， and multiple comparisons of interaction effects were also performed.
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The purpose of this paper was to introduce the orthogonal design and its quantitative data analysis of variance and the SAS implementation. From the perspective of degrees of freedom， the orthogonal design could be divided into the saturated orthogonal design and the unsaturated orthogonal design. From the perspective of the number of factor levels， the orthogonal design could be divided into the same level orthogonal design and the mixed level orthogonal design. From the perspective of normalization， the orthogonal design could also be divided into the standard orthogonal design and the non-standard orthogonal design. Quantitative data from the standard orthogonal designs could be analyzed by the conventional methods， while quantitative data from the non-standard orthogonal designs needed to be improved. Based on three examples， this paper realized the quantitative data analysis of variance with the standard orthogonal design without repeated experiments and with repeated experiments by means of the SAS software.
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The purpose of this paper was to introduce the five limitations of the PROC CAUSALGRAPH procedure and estimate the causal effect of the data by using the adjustment set based on the causal graph model. The five limitations were as follows： ①the PROC CAUSALGRAPH procedure could not deal with the causal graph model of directed circles； ② the PROC CAUSALGRAPH procedure could not evaluate dynamic processing scheme； ③ causal effect identification was a population concept； ④ causal effect identification was a nonparametric concept； ⑤ the PROC CAUSALGRAPH procedure could not identify the causal effect in some causal graph models. The example was for a simulated data set， using the conventional multiple Logistic regression model analysis and the causal graph model analysis， respectively. By comparing the analysis results of the two， the following conclusions were drawn： ① causal graph theory was useful in identifying causal effects in confounding situations； ② by implementing hierarchical estimation of causal effects， a good statistical estimation of causal effects could be achieved based on the identification results of the PROC CAUSALGRAPH procedure.
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The purpose of this paper was to introduce the methods of identifying causal effects based on instrumental variables， distinguishing different models with data， and using SAS software to realize calculation. Firstly， the four main contents of causal graph theory were introduced， including sources of association， statistical properties of causal models， identification and adjustment， and instrumental variables. Secondly， for two examples and with the help of the CAUSALGRAPH procedure in SAS/STAT， the following two tasks were completed： the first task was to identify causal effects using instrumental variables； the second task was to use data to distinguish different models.
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The purpose of this paper was to introduce the method of checking adjustment sets based on a causal graph model， finding common adjustment sets and implementing the statistical calculation with SAS software. Firstly， the basic concepts related to the causal graph model were introduced.Secondly， the primary contents of the causal graph theory were given， including the composition and terminology of the causality diagram. Finally， for the two instances and with the help of the CAUSALGRAPH procedure in SAS/STAT， the following two tasks were completed： the first task was to examine the adjustment set and enumerate paths； the second task was to find the adjustment set common to the multiple causal graph models.
ABSTRACT
The purpose of this paper was to introduce the basic knowledge of the causal graph model， the contents of the CAUSALGRAPH procedure and the method of constructing and searching adjustment sets based on the CAUSALGRAPH procedure in SAS/STAT. The causal graph model was the product of the combination of graph theory and probability theory. It could find all possible adjustment sets including the minimum adjustment set based on the action relationship between the variables set by the user. The contents of the CAUSALGRAPH procedure mainly included three identification criteria， two operating modes and one verification checking method. This paper analyzed the causal effect of two instances based on the CAUSALGRAPH procedure in SAS， and explained the output results.
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The purpose of this paper was to introduce how to set the options of variable levels and multimodal covariates， and to demonstrate the causal mediation effect analysis method with odds ratio （OR） and excess relative risk （ERR） as evaluation indicators through examples. For treatment variables， mediator variables and covariates， the variable-level options of them could be set through the evaluate statement. For categorical variables and their interaction terms， they could be treated as multimodal covariates， and the variable levels could also be set for them by using the evaluate statement. Through an example， this paper used SAS to realize the causal mediation effect analysis and the decomposition of effect components with OR and ERR as the evaluation indicators.
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The purpose of this paper was to introduce the setting method of the three types of variable levels in the causal mediation effect analysis and the implementing calculation method under the condition of stratification by using SAS. The setting of the three types of variable levels referred to the setting of the levels of treatment variable， the mediator variable and the covariate. Besides， a specific level combination could also be set for two variables. Through an example， with the help of the enveluate statement in proc causualmed procedure， this paper used an example to conduct the causal mediation effect based on different variable stratification， and gave the output results and explanations.
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The purpose of this paper was to introduce five key techniques and the multi-directional decomposition methods of effect components in the analysis of causal mediation effects. The contents of the five key technologies were as follows： ① identification of causal mediation effect； ② regression method of causal mediation effect analysis； ③ maximum likelihood estimation； ④ estimation of total effect and various component effects； ⑤ estimation by bootstrap method. The multi-directional decomposition methods included 3 bidirectional decompositions， 2 three-directional decompositions and 1 four-directional decomposition. Through an example， a causal mediation effect analysis model including covariates and interaction terms was constructed with the help of SAS， bidirectional decomposition， three-directional decomposition and four-directional decomposition were carried out for the total effect in the causal mediation effect analysis， and the output results were explained.
ABSTRACT
The purpose of this paper was to introduce the theoretical basis of the causal mediation effect analysis and the specific method to realize an example by the causal mediation effect analysis with SAS. The theoretical basis of the causal mediation effect analysis included the following two aspects， the basic concept and defining the counterfactual framework of the causal mediation effect. The example was about whether the encouraging environment provided by parents would affect the cognitive development of children. The traditional multiple linear regression analysis， the causal mediation effect analysis without considering covariates and with considering covariates were used， respectively. By comparing the results obtained by the three analysis methods， the following conclusions were drawn： ① when there were the mediation variables in the data， it was not suitable to use traditional multiple linear regression analysis to replace the causal mediation effect analysis； ② when there were covariates in the data， it was not suitable to conduct causal mediation analysis under the condition of ignoring covariates.