ANALYSIS OF VARIANCE PROCEDURE FOR ANALYZING UNBALANCED DAIRY SCIENCE DATA USING SAS

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1 ANALYSIS OF VARIANCE PROCEDURE FOR ANALYZING UNBALANCED DAIRY SCIENCE DATA USING SAS Avtar Singh National Dairy Research Institute, Karnal In statistics, analysis of variance (ANOVA) is a collection of statistical models in which the observed variance in a particular variable is partitioned into components due to different sources of variation. In its simplest form ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. There are three classes of ANOVA models: 1. Fixed-effects models assume that the data came from normal populations which may differ only in their means. 2. Random effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. 3. Mixed-effect models describe the situations where both fixed and random effects are present. In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment. One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups. The ANOVA procedure performs analysis of variance (ANOVA) for balanced data from a wide variety of experimental designs. In analysis of variance, a continuous response variable, known as a dependent variable, is measured under experimental conditions identified by classification variables, known as independent variables. The variation in the response is assumed to be due to effects in the classification, with random error accounting for the remaining variation. The ANOVA procedure is one of several procedures available in SAS software for analysis of variance. The ANOVA procedure is designed to handle balanced data (that is, data with equal numbers of observations for every combination of the classification factors), whereas the GLM procedure can analyze both balanced and unbalanced data. If your design does not pass the test, PROC ANOVA produces a warning message to tell you that the design is unbalanced and that the ANOVA analyses might not be valid. If you re unsure about the validity of PROC ANOVA for your design, you should use PROC GLM. The ANOVA procedure automatically produces graphics as part of its ODS (Output Delivery System) output. Effective graphics are indispensable for modern statistical analysis. They reveal patterns, differences, and uncertainty that are not readily apparent in tabular output. Graphics provoke questions that stimulate deeper investigation, and they add visual clarity and rich content to reports and presentations. With ODS Graphics, statistical procedures now produce graphs as automatically as they produce tables, and graphs are now integrated with tables in the ODS output. ODS Graphics is enabled when you specify the following statement: ods graphics on;

2 When ODS Graphics is enabled, procedures that support ODS Graphics create appropriate graphs, either by default or when you specify procedure options for requesting specific graphs. These options are documented in the "Syntax" section of each procedure chapter, and the "Details" section of each chapter provides an "ODS Graphics" subsection that lists the graphs that are available. Once ODS Graphics is enabled, it stays enabled for the duration of your SAS session. Alternatively, you can turn it off as follows: ods graphics off; For example, you might consider disabling ODS Graphics if your goal is solely to produce computational results. Often though, you can specify ODS GRAPHICS ON and then leave it on. APPLICATION OF PR0C GLM FOR UNBALANCED DATA Analysis of variance, or ANOVA, typically refers to partitioning the variation in a variable s values into variation between and within several groups or classes of observations. The GLM procedure can perform simple or complicated ANOVA for balanced or unbalanced data. This example discusses the analysis of variance for the unbalanced data shown below. The experimental design is a full factorial, in which each level of one treatment factor occurs at each level of the other treatment factor. Note that there is only one value for the cell with A= A2 and B= B2. Since one cell contains a different number of values from the other cells in the table, this is an unbalanced design. Unbalanced Two-Way Data A1 A2 B1 12, 14 20, 18 B2 11, 9 17 The following statements read the data into a SAS data set and then invoke PROC GLM to produce the analysis. Sr. No Month Year Y305 Sr. No Month Year Y305 Sr. No Month Year Y Jan Dec Jan May Dec Feb April Feb April Jun Oct Feb Jan April May Sept Feb Jan Feb April July April Aug Jan Jan Jan Nov Jan Jan Jan Feb Mar Jan Jan Mar Jan Feb April Jan May Feb Nov Feb Jan April April Feb April Aug Mar Mar Feb Jul Jun

3 38. Dec Dec Mar Mar Feb Mar Jan Mar Aug Mar April Nov Aug Mar April Mar Jan May Jun Mar Feb Mar May Mar Sept Aug July Jun Sept Oct Dec Oct Jan Feb Mar Jun Aug Jun July Feb Dec Jun Feb Jan Mar Feb May Jun Mar Nov Jan Sept Jan April Mar Sept Feb SAS Procedure title 'Analysis of Unbalanced 2-by-2 Factorial'; data exp; input A $ B $ datalines; A1 B1 12 A1 B1 14 A1 B2 11 A1 B2 9 A2 B1 20 A2 B1 18 A2 B2 17 ; proc glm data=exp; class A B; model Y=A B A*B; run; KARAN FRIES DATA ANALYSIS USING TWO-WAY INTERACTION Now we will analyze the sample data of Karan Fries cattle for 305-day milk yield taking month and year as factors affecting milk yield. SAS Output Class Levels Values Year Month 12 april aug dec feb jan jul jun mar may nov oct sept Number of Observations Read Number of Observations Used 112

4 Dependent Variable: Y305 Y305 Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE Y305 Mean Source DF Type I SS Mean Square F Value Pr > F Year Month Year*Month Source DF Type III SS Mean Square F Value Pr > F Year Month Year*Month Level of Y Year N Mean Std Dev Level of Y Month N Mean Std Dev April Aug Dec Feb Jan Jul Jun Mar May Nov Oct Sept

Least Squares MEAN Year Y305 LSMEAN Number 90 3056.

5 Least Squares MEAN Year Y305 LSMEAN Number

7 MIXED MODEL ANALYSIS SAS Procedure title 'Expected Mean Squares for the Correct Model'; proc mixed data=sasuser.scores method=type3; class material teacher; model score=material; random teacher(material); run; title 'Expected Mean Squares for the Incorrect Model'; proc mixed data=sasuser.scores method=type3; class material teacher; model score=material teacher(material); run; SAS Output: 1. Expected Mean Squares for the Correct Model Type 3 Analysis of Variance Sum of Source DF Squares Mean Square Expected Mean Square Material Var(Residual) + 6 Var(Teacher(Material)) + Q(Material) Teacher(Material) Var(Residual) + 6 Var(Teacher(Material)) Residual Var(Residual) Covariance Parameter Estimates Cov Parm Estimate Teacher(Material) Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Material

8 2. Expected Mean Squares for the Incorrect Model Type 3 Analysis of Variance Sum of Source DF Squares Mean Square Expected Mean Square Material Var(Residual) + Q(Material,Teacher(Material)) Teacher(Material) Var(Residual) + Q(Teacher(Material)) Residual Var(Residual) Covariance Parameter Estimates Cov Parm Estimate Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Material Teacher(Material) <.0001 Follow up tests A statistically significant effect in ANOVA is often followed up with one or more different follow-up tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow up tests are often distinguished in terms of whether they are planned (a priori) or post hoc. Planned tests are determined before looking at the data and post hoc tests are performed after looking at the data. Post hoc tests such as Tukey's range test most commonly compare every group mean with every other group mean and typically incorporate some method of controlling for Type I errors. Comparisons, which are most commonly planned, can be either simple or compound. Simple comparisons compare one group mean with one other group mean. Compound comparisons typically compare two sets of groups means where one set has at two or more groups (e.g., compare average group means of group A, B and C with group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels.

Two-Factor unbalanced experiment with factors of Power and Humidity Example compares LSmeans and means statement for unbalanced data

STAT:5201 Anaylsis/Applied Statistic II (LSmeans vs. means) Two-Factor unbalanced experiment with factors of Power and Humidity Example compares LSmeans and means statement for unbalanced data Power (levels