/* Two-way ANOVA with interaction: Unequal sample sizes */ OPTIONS NODATE LS=77 PS=55 PAGENO=1; DATA one; INPUT growth gender $ bone $ ; DATAlines; 1.4 Male Severe 2.4 Male Severe 2.2 Male Severe 2.1 Male Moderate 1.7 Male Moderate 0.7 Male Mild 1.1 Male Mild 2.4 Female Severe 2.5 Female Moderate 1.8 Female Moderate 2.0 Female Moderate 0.5 Female Mild 0.9 Female Mild 1.3 Female Mild ; PROC TABULATE DATA=one; title 'Table of Cell Means'; CLASS gender bone / DESCENDING; VAR growth; TABLE (gender all), growth*(bone all)*(N*f=4.0 MEAN*f=8.4); RUN; PROC GLM; title 'Two-way ANOVA: growth versus bone'; CLASS bone; MODEL growth = bone; RUN; PROC GLM; title 'Two-way ANOVA: growth versus gender and bone'; CLASS gender bone; MODEL growth = gender bone; RUN; PROC GLM; title 'Two-way ANOVA: growth versus gender and bone and interaction'; CLASS gender bone; MODEL growth = gender bone gender*bone; means gender bone / t; lsmeans gender bone / t; RUN; DATA two; INPUT growth gender1 bone1 bone2; gb1=gender1*bone1; gb2=gender1*bone2; DATALINES; 1.4 1 1 0 2.4 1 1 0 2.2 1 1 0 2.1 1 0 1 1.7 1 0 1 0.7 1 -1 -1 1.1 1 -1 -1 2.4 -1 1 0 2.5 -1 0 1 1.8 -1 0 1 2.0 -1 0 1 0.5 -1 -1 -1 0.9 -1 -1 -1 1.3 -1 -1 -1 ; PROC PRINT; title 'Data for regression approach'; PROC REG; TITLE 'Row Effects'; MODEL growth = bone1 bone2; RUN; PROC REG; TITLE 'Row and Column Effects'; MODEL growth = gender1 bone1 bone2; RUN; PROC REG; TITLE 'Row, Column, and Interaction Effects'; MODEL growth = gender1 bone1 bone2 gb1 gb2; MODEL growth = gender1 bone1 bone2 gb1 gb2 / SELECTION=RSQUARE SSE; RUN; QUIT; ***************************************************************************** Table of Cell Means 1 --------------------------------------------------------------------------- growth --------------------------------------------------------- bone ------------------------------------------- Severe Moderate Mild All --------------------------------------------------------- N Mean N Mean N Mean N Mean --------------------------------------------------------------------------- gender ------------------- Male 3 2.0000 2 1.9000 2 0.9000 7 1.6571 --------------------------------------------------------------------------- Female 1 2.4000 3 2.1000 3 0.9000 7 1.6286 --------------------------------------------------------------------------- All 4 2.1000 5 2.0200 5 0.9000 14 1.6429 --------------------------------------------------------------------------- Two-way ANOVA: growth versus bone 2 The GLM Procedure Class Level Information Class Levels Values bone 3 Mild Moderate Severe Number of Observations Read 14 Number of Observations Used 14 Two-way ANOVA: growth versus bone 3 The GLM Procedure Dependent Variable: growth Sum of Source DF Squares Mean Square F Value Pr > F Model 2 4.30628571 2.15314286 16.13 0.0005 Error 11 1.46800000 0.13345455 Corrected Total 13 5.77428571 R-Square Coeff Var Root MSE growth Mean 0.745769 22.23652 0.365314 1.642857 Source DF Type I SS Mean Square F Value Pr > F bone 2 4.30628571 2.15314286 16.13 0.0005 Source DF Type III SS Mean Square F Value Pr > F bone 2 4.30628571 2.15314286 16.13 0.0005 Two-way ANOVA: growth versus gender and bone 4 The GLM Procedure Class Level Information Class Levels Values gender 2 Female Male bone 3 Mild Moderate Severe Number of Observations Read 14 Number of Observations Used 14 Two-way ANOVA: growth versus gender and bone 5 The GLM Procedure Dependent Variable: growth Sum of Source DF Squares Mean Square F Value Pr > F Model 3 4.39885714 1.46628571 10.66 0.0019 Error 10 1.37542857 0.13754286 Corrected Total 13 5.77428571 R-Square Coeff Var Root MSE growth Mean 0.761801 22.57456 0.370868 1.642857 Source DF Type I SS Mean Square F Value Pr > F gender 1 0.00285714 0.00285714 0.02 0.8883 bone 2 4.39600000 2.19800000 15.98 0.0008 Source DF Type III SS Mean Square F Value Pr > F gender 1 0.09257143 0.09257143 0.67 0.4311 bone 2 4.39600000 2.19800000 15.98 0.0008 Two-way ANOVA: growth versus gender and bone and interaction 6 The GLM Procedure Class Level Information Class Levels Values gender 2 Female Male bone 3 Mild Moderate Severe Number of Observations Read 14 Number of Observations Used 14 Two-way ANOVA: growth versus gender and bone and interaction 7 The GLM Procedure Dependent Variable: growth Sum of Source DF Squares Mean Square F Value Pr > F Model 5 4.47428571 0.89485714 5.51 0.0172 Error 8 1.30000000 0.16250000 Corrected Total 13 5.77428571 R-Square Coeff Var Root MSE growth Mean 0.774864 24.53731 0.403113 1.642857 Source DF Type I SS Mean Square F Value Pr > F gender 1 0.00285714 0.00285714 0.02 0.8978 bone 2 4.39600000 2.19800000 13.53 0.0027 gender*bone 2 0.07542857 0.03771429 0.23 0.7980 Source DF Type III SS Mean Square F Value Pr > F gender 1 0.12000000 0.12000000 0.74 0.4152 bone 2 4.18971429 2.09485714 12.89 0.0031 gender*bone 2 0.07542857 0.03771429 0.23 0.7980 Two-way ANOVA: growth versus gender and bone and interaction 8 The GLM Procedure t Tests (LSD) for growth NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 8 Error Mean Square 0.1625 Critical Value of t 2.30600 Least Significant Difference 0.4969 Means with the same letter are not significantly different. t Grouping Mean N gender A 1.6571 7 Male A A 1.6286 7 Female Two-way ANOVA: growth versus gender and bone and interaction 9 The GLM Procedure t Tests (LSD) for growth NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 8 Error Mean Square 0.1625 Critical Value of t 2.30600 Comparisons significant at the 0.05 level are indicated by ***. Difference bone Between 95% Confidence Comparison Means Limits Severe - Moderate 0.0800 -0.5436 0.7036 Severe - Mild 1.2000 0.5764 1.8236 *** Moderate - Severe -0.0800 -0.7036 0.5436 Moderate - Mild 1.1200 0.5321 1.7079 *** Mild - Severe -1.2000 -1.8236 -0.5764 *** Mild - Moderate -1.1200 -1.7079 -0.5321 *** Two-way ANOVA: growth versus gender and bone and interaction 10 The GLM Procedure Least Squares Means growth H0:LSMean1=LSMean2 gender LSMEAN t Value Pr > |t| Female 1.80000000 0.86 0.4152 Male 1.60000000 growth LSMEAN bone LSMEAN Number Mild 0.90000000 1 Moderate 2.00000000 2 Severe 2.20000000 3 Least Squares Means for Effect bone t for H0: LSMean(i)=LSMean(j) / Pr > |t| Dependent Variable: growth i/j 1 2 3 1 -4.22738 -4.38178 0.0029 0.0023 2 4.227383 -0.67412 0.0029 0.5192 3 4.38178 0.67412 0.0023 0.5192 NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. Data for regression approach 11 Obs growth gender1 bone1 bone2 gb1 gb2 1 1.4 1 1 0 1 0 2 2.4 1 1 0 1 0 3 2.2 1 1 0 1 0 4 2.1 1 0 1 0 1 5 1.7 1 0 1 0 1 6 0.7 1 -1 -1 -1 -1 7 1.1 1 -1 -1 -1 -1 8 2.4 -1 1 0 -1 0 9 2.5 -1 0 1 0 -1 10 1.8 -1 0 1 0 -1 11 2.0 -1 0 1 0 -1 12 0.5 -1 -1 -1 1 1 13 0.9 -1 -1 -1 1 1 14 1.3 -1 -1 -1 1 1 Row Effects 12 The REG Procedure Model: MODEL1 Dependent Variable: growth Number of Observations Read 14 Number of Observations Used 14 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 4.30629 2.15314 16.13 0.0005 Error 11 1.46800 0.13345 Corrected Total 13 5.77429 Root MSE 0.36531 R-Square 0.7458 Dependent Mean 1.64286 Adj R-Sq 0.6995 Coeff Var 22.23652 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.67333 0.09818 17.04 <.0001 bone1 1 0.42667 0.14408 2.96 0.0129 bone2 1 0.34667 0.13614 2.55 0.0272 Row and Column Effects 13 The REG Procedure Model: MODEL1 Dependent Variable: growth Number of Observations Read 14 Number of Observations Used 14 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 4.39886 1.46629 10.66 0.0019 Error 10 1.37543 0.13754 Corrected Total 13 5.77429 Root MSE 0.37087 R-Square 0.7618 Dependent Mean 1.64286 Adj R-Sq 0.6903 Coeff Var 22.57456 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.67619 0.09973 16.81 <.0001 gender1 1 -0.08571 0.10448 -0.82 0.4311 bone1 1 0.46667 0.15418 3.03 0.0127 bone2 1 0.32667 0.14035 2.33 0.0422 Row, Column, and Interaction Effects 14 The REG Procedure Model: MODEL1 Dependent Variable: growth Number of Observations Read 14 Number of Observations Used 14 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 4.47429 0.89486 5.51 0.0172 Error 8 1.30000 0.16250 Corrected Total 13 5.77429 Root MSE 0.40311 R-Square 0.7749 Dependent Mean 1.64286 Adj R-Sq 0.6342 Coeff Var 24.53731 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.70000 0.11637 14.61 <.0001 gender1 1 -0.10000 0.11637 -0.86 0.4152 bone1 1 0.50000 0.17776 2.81 0.0227 bone2 1 0.30000 0.15756 1.90 0.0934 gb1 1 -0.10000 0.17776 -0.56 0.5891 gb2 1 -2.5443E-17 0.15756 -0.00 1.0000 Row, Column, and Interaction Effects 15 The REG Procedure Model: MODEL2 Dependent Variable: growth R-Square Selection Method Number of Observations Read 14 Number of Observations Used 14 Number in Model R-Square SSE Variables in Model 1 0.5959 2.33328 bone1 1 0.5431 2.63829 bone2 1 0.0443 5.51829 gb2 1 0.0038 5.75231 gb1 1 0.0005 5.77143 gender1 -------------------------------------------------------------------- 2 0.7458 1.46800 bone1 bone2 2 0.6328 2.12057 gender1 bone1 2 0.6255 2.16262 bone1 gb1 2 0.6127 2.23631 bone1 gb2 2 0.5473 2.61429 bone2 gb2 2 0.5436 2.63543 gender1 bone2 2 0.5435 2.63619 bone2 gb1 2 0.0484 5.49473 gb1 gb2 2 0.0448 5.51543 gender1 gb2 2 0.0041 5.75071 gender1 gb1 -------------------------------------------------------------------- 3 0.7618 1.37543 gender1 bone1 bone2 3 0.7538 1.42185 bone1 bone2 gb1 3 0.7499 1.44400 bone1 bone2 gb2 3 0.6728 1.88963 gender1 bone1 gb1 3 0.6481 2.03185 gender1 bone1 gb2 3 0.6269 2.15413 bone1 gb1 gb2 3 0.5513 2.59073 bone2 gb1 gb2 3 0.5477 2.61143 gender1 bone2 gb2 3 0.5440 2.63283 gender1 bone2 gb1 3 0.0493 5.48971 gender1 gb1 gb2 -------------------------------------------------------------------- 4 0.7749 1.30000 gender1 bone1 bone2 gb1 4 0.7660 1.35143 gender1 bone1 bone2 gb2 4 0.7541 1.42000 bone1 bone2 gb1 gb2 4 0.6728 1.88909 gender1 bone1 gb1 gb2 4 0.5522 2.58571 gender1 bone2 gb1 gb2 -------------------------------------------------------------------- 5 0.7749 1.30000 gender1 bone1 bone2 gb1 gb2