ANALYSIS OF VARIANCE OF BALANCED DAIRY SCIENCE DATA USING SAS Ravinder Malhotra and Vipul Sharma National Dairy Research Institute, Karnal-132001 The most common use of statistics in dairy science is testing hypothesis difference between two or more categorical treatments group. When we compared means from k independent groups, where k is greater than 2, the technique is called Analysis of Variance (ANOVA).It is a powerful statistics tool for test of significance. It is a family of methods that can be used to design and analyze the results from both simple and complex experiments. It is one of the most important statistical techniques available to researchers and provides a link between the design of experiments and the analysis of experimental data. In ANOVA the total variation in the sample data is expressed as the sum of its non negative components where each of these components is a measure of variation due to some specific independent source. 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. Balanced data is that data with equal numbers of observations for every combination of the classification factors. One Way ANOVA A one-way analysis of variance is an extension of the independent group t-test where there are more than two groups. A one-way analysis of variance considers one treatment factor with two or more treatment levels. The goal of the analysis is to test for differences among the means of the levels and to quantify these differences. For example in milk filling plant, four machines are filling milk in the cans to a level of 12.0 litres. In order to test the differences among machine with regard to average quantity of milk filled in the cans we can perform one way ANOVA. Assumptions: It is assumed that subjects are randomly assigned to one of 3 or more groups and that the data within each group are normally distributed with equal variances across groups. Sample sizes between groups do not have to be equal, but large differences in sample sizes for the groups may affect the outcome of some multiple comparisons tests. Test: The hypotheses for the comparison of independent groups are: H0 1 2... k (means of all the groups are equal) k is the number of groups H a i j (means of the two or more groups are not equal) The test statistic reported is an F-test with k-1 and N-k degrees of freedom, where N is the number of subjects. A low p-value for the F-test is evidence to reject the null hypothesis. In other words, there is evidence that at least one pair of means are not equal. PROC ANOVA and PROC GLM are two procedures in SAS designed for analyzing ANOVA models. The general form of PROC ANOVA is, PROC ANOVA data = data set; CLASS variables; /* Identify the variable that divide the data set into groups. MODEL dependent variable = independent variable; MEANS variables / options RUN;
The general form of PROC GLM is, PROC GLM data = data set; CLASS variables; /* Identify the variable that divide the data set into groups. MODEL dependent variable = independent variable; MEANS variables / options RUN; The MEANS statement requests multiple comparisons for the class variable listed after MEANS. Include a MEANS statement with a multiple comparison option. The syntax for this statement is MEANS variable name /test name; where test name is a multiple comparison test. Some of the tests available in SAS include BON - Performs Bonferroni t-tests of differences DUNCAN - Duncan s multiple range test SCHEFFE - Scheffe multiple comparison procedure SNK - Student Newman Keuls multiple range test LSD - Fisher s Least Significant Difference test TUKEY - Tukey s studentized range test DUNNETT - Dunnett s test compare to a single control specify For example, to select the TUKEY test, we would use the statement CASE STUDY MEANS GROUP /TUKEY; Procedure is discussed in Example 1. Two-way ANOVA Two-way ANOVA is used to compare the means of populations that are classified in two different ways, or the mean responses in an experiment with two factors. PROC GLM and PROC ANOVA are two procedures in SAS designed for analyzing ANOVA models. Only simple modifications are needed to incorporate the additional factor and possible interactions between the factors. The general form of PROC GLM is, PROC GLM data = data set; CLASS variables; /* Identify the variables that divide the data set into groups. MODEL response variable = explanatory variables; MEANS variables / options RUN; In the program above the CLASS statement designates which variables are factors in the model. For example, the statement: CLASS A B tells SAS that the variables A and B are factors in the ANOVA model. The MODEL statement is used to define the format of the model. In two-way ANOVA both factors should be included in this statement. In addition, the interaction term should be included here if you want one to be part of the model. Assuming that y specifies the response variable, the following statement specifies a two-way ANOVA model without an interaction term is: MODEL y = A B (procedure is discussed in Example 2) The statement MODEL y = A B A*B specifies a two-way ANOVA model with interaction; This can alternatively be written in short-hand form as: MODEL y = A B; (Procedure is discussed in Example 3)
Both PROC GLM and PROC ANOVA are procedures that can be used for analyzing ANOVA models. PROC GLM is an all-purpose procedure that can be used to analyze all types of general linear models (including ANOVA, regression and multivariate models). Because PROC GLM will do so many things, it generally uses more computer resources to generate the same output as PROC ANOVA. However, PROC ANOVA is not appropriate to use when the data are unbalanced. It is also not able to create the contrasts that may be needed in a follow-up analysis. Therefore it is recommended to use PROC GLM when performing ANOVA in SAS. Types of Sums of Squares for ANOVA Effects a) Type I SS are order-dependent (hierarchical, sequential). Each effect is adjusted for all other effects that appear earlier (to the left) in the model, but not for any effects that appear later in the model. It is appropriate for balanced (orthogonal, equal n), analyses of variance in which the effects are specified in proper order (main effects, then two-way interactions, then three-way interactions, etc.) and for trend analysis where the powers for the quantitative factor are ordered from lowest to highest in the model statement. b) Type II SS involve the adjustments regardless of the order of the effects in the model statement. There is reduction in the SSE due to adding the effect to a model that contains all other effects except effects that contain the effect being tested.. c) Type III SS are each adjusted for all other effects in the model, regardless of order. It is recommended in non-orthogonal data. This is also the SS that are approximated by the traditional unweighted means ANOVA that uses harmonic mean sample sizes to adjust cell totals. d) Type IV SS are identical to Type III SS for designs with no missing cells. Multiple Comparison Procedures a) Least Significant Difference (LSD) The aim of the procedure is to determine the least difference between a pair of means that will be significant and to compare that value with the calculated differences between two means is greater than the least significant difference, it can be concluded that the difference between this pair of means is significant. LSD is computed LSD t ij / 2, N MS RES 1 ni 1 n j where MS RES ni 1 n j 1 is the standard error of the estimator of the difference between the mean of two groups i and j. b) Scheffe' Test A test used to find where the differences between means lie when the Analysis of Variance indicates the means are not all equal. The Scheffe' test is generally used when the sample sizes are different. It uses a critical value from ANOVA and multiply by k- 1where k is number of groups (means) F' critical = (k 1) F critical xi x j ' Test Value Fs If F s F Critical then the two means are significantly different 2 S 1 1 w ni n j c) Tukey s Test A test used to find where the differences between the means lie when the Analysis of Variance indicates the means are not all equal. The Tukey test is generally used when the sample sizes are all the same. All means for each condition are ranked in order of magnitude; group with lowest mean gets a ranking of 1. The pairwise differences between means, starting with the largest mean compared to the smallest mean, are tabulated
between each group pair and divided by the standard error. The value, q computed xi x j as q, is compared to a Studentized range critical value. If q is larger than the 2 sw / n critical value, then the expression between that group pair is considered to be statistically different. d) Student-Newman-Keuls (SNK) test: This test is similar to the Tukey test, except with regard to how the critical value is determined. All q s in Tukey s test are compared to the same critical value determined for that experiment; whereas all q s determined from SNK test are compared to a different critical value. This makes the SNK test slightly less conservative than the Tukey test. Tests for Homogeneity of Variances a) Levene s Test b) Brown-Forsythe Test c) F max Test Checking for Normality If the assumptions for the ANOVA hold, the values from each sample should come from a normal distribution. Departures from normality can suggest the presence of outliers in the data, or of a non-normal distribution in one or more of the samples. The normality test will give an indication of whether the populations from which the samples were drawn appear to be normally distributed, but will not indicate the cause(s) of the non-normality. The smaller the sample size, the less likely the normality test will be able to detect non-normality. Normality tests: a) Kolmogorov-Smirnov test; Anderson-Darling test (both based on the empirical CDF). b) Shapiro-Wilk s test; Ryan-Joiner test (both are correlation based applicable for n < 50). c) D Agostino s test (n>=50). Graphical methods: The graphic displays can convey the patterns and relationships easily than by other analytic methods. However, the power of graphical methods relies on our eye-brain system and the graphical technique a) Histograms: The histogram each sample has a reference normal distribution curve for a normal distribution with the same mean and variance as the sample. This provides a reference for detecting gross non-normality when the sample sizes are large. b) Normality test for residuals: If the assumptions for the ANOVA hold, all the residuals should come from the same normal distribution with mean 0. Departures from normality can suggest the presence of outliers in the data, or of a non-normal distribution in one or more of the populations from which the samples were drawn. c) Histogram for residuals: The histogram for residuals has a reference normal distribution curve for a normal distribution with the same mean and variance as the residuals. This provides a reference for detecting gross nonnormality when the sample sizes are large. d) Boxplot for residuals: Suspected outliers appear in a boxplot as individual points o or x outside the box. If these appear on both sides of the box, they also suggest the possibility of a heavy-tailed distribution. If they appear on only one side, they also suggest the possibility of a skewed distribution. Skewness is also suggested if the mean (+) does not lie on or near the central line of the boxplot, or if the central line of the boxplot does not evenly divide the box.
e) Normal probability plot for residuals: For data sampled from a normal distribution, the normal probability plot, (normal Q-Q plot) has the points all lying on or near the straight line drawn through the middle half of the points. Scattered points lying away from the line are suspected outliers. f) Residuals plotted against fitted values: If the fitted model under the assumption of populations with equal variance is correct, the plot of residuals against fitted values should suggest a horizontal band across the graph. The graph of residuals against fitted values will consist of vertical "stacks" of residuals, one stack for each unique sample mean. The stacks should be about the same length and at about the same level. Outliers may appear as anomalous points in the graph (although an outlier may not turn up in the residuals plot by virtue of affecting the mean so that its fitted value lies near it). g) Kernel Density Plot: Another graphical method for normality test is the kernel density plot that portrays the distribution of data directly. In order to get the plot, we first have to perform statistical density estimation, which involves approximating a hypothesized probability density function from the observed data. Kernel density estimation is a nonparametric technique for density estimation in which a known density function (kernel) is averaged across the observed data points to create a smooth approximation. After plotting the density function, we can easily check the normality by comparing the shape of resulting plot with the bell-shaped curve of normal distribution. Example 1: In milk filling plant, four machines are filling milk in the cans to a level of 12.0 litres random samples of the cans from each of the four machines were taken. The following table gives the quantity of milk filled in the cans by different machines. Machine Quantity of Milk A 11.95 A 12.05 A 12.1 A 12.3 A. B 12.6 B 12.09 B 12.12 B 11.95 B 12.05 C 11.9 C 11.95 C 11.92 C 11.85 C 11.88 D 12.1 D 12.15 D 12.05 D. D. Test the differences among machine with regard to average quantity of milk filled in the cans
SAS Procedure PROC ANOVA DATA=WORK.TMP0TempTableInput ; CLASS Machine; MODEL "Qunatity of Milk"n = Machine ; MEANS Machine / TUKEY ALPHA=0.05 ; MEANS Machine / LSD ALPHA=0.05 ; RUN; QUIT; One-Way Analysis of Variance Results Class Level Information Class Levels Values Machine 4 A B C D Number of Observations Read 20 Number of Observations Used 17 Source DF Sum of Squares Mean Square F Value Pr > F Model 3 0.19121412 0.06373804 2.50 0.1057 Error 13 0.33208000 0.02554462 Corrected Total 16 0.52329412 R Square Coeff Var Root MSE Quantity of Milk Mean 0.365405 1.325329 0.159827 12.05941 Source DF Anova SS Mean Square F Value Pr > F Machine 3 0.19121412 0.06373804 2.50 0.1057
Tukey's Studentized Range (HSD) Test for Quantity of Milk Alpha 0.05 Error Degrees of Freedom 13 Error Mean Square 0.025545 Critical Value of Studentized Range 4.15087 Comparisons significant at the 0.05 level are indicated by ***. Difference Between Simultaneous 95% Means Confidence Limits Machine Comparison B A 0.0620 0.2527 0.3767 B D 0.0620 0.2806 0.4046 B C 0.2620 0.0347 0.5587 A B 0.0620 0.3767 0.2527 A D 0.0000 0.3583 0.3583 A C 0.2000 0.1147 0.5147 D B 0.0620 0.4046 0.2806 D A 0.0000 0.3583 0.3583 D C 0.2000 0.1426 0.5426 C B 0.2620 0.5587 0.0347 C A 0.2000 0.5147 0.1147 C D 0.2000 0.5426 0.1426 t Tests (LSD) for Quantity of Milk Alpha 0.05 Error Degrees of Freedom 13 Error Mean Square 0.025545 Critical Value of t 2.16037 Comparisons significant at the 0.05 level are indicated by ***. Machine Comparison Difference Between Means 95% Confidence Limits B A 0.0620 0.1696 0.2936 B D 0.0620 0.1902 0.3142 B C 0.2620 0.0436 0.4804 *** A B 0.0620 0.2936 0.1696 A D 0.0000 0.2637 0.2637 A C 0.2000 0.0316 0.4316 D B 0.0620 0.3142 0.1902 D A 0.0000 0.2637 0.2637 D C 0.2000 0.0522 0.4522 C B 0.2620 0.4804 0.0436 *** C A 0.2000 0.4316 0.0316 C D 0.2000 0.4522 0.0522 Means Plot of 'Quantity of Milk' by Machine
Example 2: An experiment was conducted with Five levels of Total Solid (17 %, 20%, 23% 26% and 30 %) to examine the influence on the quality of ice-cream.a sensory evaluation was carried out and the table below gives the overall acceptability scores of Ice cream prepared with 5 levels of total solids given by a panel of seven judges. Obs ratio judge score 1 1 1 5.72 2 1 2 5.48 3 1 3 5.78 4 1 4 5.68 5 1 5 5.48 6 1 6 5.44 7 1 7 5.45 8 2 1 5.58 9 2 2 5.48 10 2 3 5.78 11 2 4 5.68 12 2 5 5.48 13 2 6 5.44 14 2 7 5.45 15 3 1 7.54 16 3 2 7.59 17 3 3 7.68 18 3 4 7.95 19 3 5 7.84 20 3 6 7.68 21 3 7 7.87 22 4 1 7.80 23 4 2 7.98 24 4 3 7.89 25 4 4 7.87 26 4 5 7.89 27 4 6 7.98 28 4 7 7.86 29 5 1 4.32 30 5 2 5.00 31 5 3 5.36 32 5 4 4.98 33 5 5 4.86
34 5 6 4.65 35 5 7 4.86 SAS Procedure proc glm data=rmsas.two_way_one_obs plots(only)=diagnostics(unpack); class fat_ratio judge; model score=fat_ratio judge; title 'TWO WAY ANOVA for Consistency Score'; run; quit; TWO WAY ANOVA for Consistency Score The GLM Procedure Source DF Sum of Squares Mean Square F Value Pr > F Model 10 54.60745714 5.46074571 194.39 <.0001 Error 24 0.67421714 0.02809238 Corrected Total 34 55.28167429 R Square Coeff Var Root MSE score Mean 0.987804 2.649986 0.167608 6.324857 Source DF Type I SS Mean Square F Value Pr > F fat_ratio 4 54.26770286 13.56692571 482.94 <.0001 judge 6 0.33975429 0.05662571 2.02 0.1029 Source DF Type III SS Mean Square F Value Pr > F fat_ratio 4 54.26770286 13.56692571 482.94 <.0001 judge 6 0.33975429 0.05662571 2.02 0.1029
Example 3: Four batches of paneer were prepared by taking three levels of fat contents of milk and three churning speeds and moisture contents of these samples were recorded and are given below : SAS Procedure fat_ moisture_ Obs level speed batch content 1 1 1 1 66.8 2 1 2 1 69.1 3 1 3 1 69.3 4 2 1 1 77.9 5 2 2 1 77.8 6 2 3 1 76.1 7 3 1 1 71.7 8 3 2 1 72.2 9 3 3 1 71.9 10 1 1 2 68.1 11 1 2 2 68.9 12 1 3 2 70.1 13 2 1 2 77.6 14 2 2 2 78.7 15 2 3 2 76.4 16 3 1 2 72.3 17 3 2 2 71.9 18 3 3 2 71.8 19 1 1 3 67.0 20 1 2 3 69.2 21 1 3 3 70.0 22 2 1 3 77.0 23 2 2 3 78.2 24 2 3 3 76.0 25 3 1 3 72.2 26 3 2 3 72.3 27 3 3 3 71.8 28 1 1 4 67.5 29 1 2 4 68.1 30 1 3 4 68.9 31 2 1 4 77.1 32 2 2 4 77.6 33 2 3 4 75.7 34 3 1 4 71.5 35 3 2 4 71.4 36 3 3 4 71.3 proc glm data=rmsas.two_way_inter; class fat_level speed batch; model moisture_content=fat_level speed fat_level*speed; title 'Analyze the Effects of fat_level & speed'; title2 'Including Interaction'; run; quit;
Analyze the Effects of fat_level & speed Including Interaction The GLM Procedure Source DF Sum of Squares Mean Square F Value Pr > F Model 8 470.1455556 58.7681944 293.84 <.0001 Error 27 5.4000000 0.2000000 Corrected Total 35 475.5455556 R Square Coeff Var Root MSE moisture_content Mean 0.988645 0.616516 0.447214 72.53889 Source DF Type I SS Mean Square F Value Pr > F fat_level 2 451.2372222 225.6186111 1128.09 <.0001 speed 2 3.3238889 1.6619444 8.31 0.0015 fat_level*speed 4 15.5844444 3.8961111 19.48 <.0001 Source DF Type III SS Mean Square F Value Pr > F fat_level 2 451.2372222 225.6186111 1128.09 <.0001 speed 2 3.3238889 1.6619444 8.31 0.0015 fat_level*speed 4 15.5844444 3.8961111 19.48 <.0001 References: Susan J. Slaughter & Lora D. Delwiche. The Little SAS Book for Enterprise Guide 4.2. Kaps M and Lamberson W R. Bio Statistics for Animal Sciences, CABI Publications http://support.sas.com/