GENERAL LINEAR MODELS (GLM) - NDSU · 2019. 10. 29. · GENERAL LINEAR MODELS (GLM) • The GLM...
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GENERAL LINEAR MODELS (GLM) • The GLM method allows for performing analysis of variance of balanced or unbalanced data using analysis of variance (ANOVA). • GLM uses a general linear model method for performing the ANOVA. • The GLM method calculates Type I and Type III sums of squares. • The GLM method allows for performing analyses using a mixed model • Many journals are requiring that papers submitted to their journals be analyzed using a mixed model method, such as PROC MIXED, instead of GLM. • Many of the mixed model methods analyze the data using a mathematical method called restricted maximum likelihood (REML). • If you have no missing data and the data are balanced, the GLM and REML methods provide nearly identical results. TYPE I vs. TYPE III Sum of Squares Type I SS • Referred to as the Sequential or Hierarchical Sum of Squares. • The Type I SS should be used only if the data are balanced, including no missing data. • The order the terms appear in the model is important in the sum of square value calculated. • The Sum of Square calculated is adjusted for the term(s) that appear before it in the model statement. • For example, if the model is yield=A B A*B, the Type I SS are calculated as o SS A o SS B|A (SS B given A is in the model o SS A*B|A, B (SS A*B given A and B are in the model • THESE SUM SQUARES SHOULD NOT BE USED FOR HYPOTHESIS TESTING UNLESS YOU HAVE BALANCED DATA, INCLUDING NO MISSING DATA. Type III SS • THESE SUMS OF SQUARE ARE THE APPROPRIATE ONES TO USE FOR F- TESTS IF YOU HAVE UNBALANCED DATA, INCLUDING MISSING DATA. • The order of terms in the model statement does not affect the calculated sums of squares.
Transcript of GENERAL LINEAR MODELS (GLM) - NDSU · 2019. 10. 29. · GENERAL LINEAR MODELS (GLM) • The GLM...
GENERAL LINEAR MODELS (GLM)
• The GLM method allows for performing analysis of variance of balanced or unbalanced data using analysis of variance (ANOVA).
• GLM uses a general linear model method for performing the ANOVA.
• The GLM method calculates Type I and Type III sums of squares.
• The GLM method allows for performing analyses using a mixed model
• Many journals are requiring that papers submitted to their journals be analyzed using a mixed model method, such as PROC MIXED, instead of GLM.
• Many of the mixed model methods analyze the data using a mathematical method called restricted maximum likelihood (REML).
• If you have no missing data and the data are balanced, the GLM and REML methods provide nearly identical results.
TYPE I vs. TYPE III Sum of Squares Type I SS
• Referred to as the Sequential or Hierarchical Sum of Squares.
• The Type I SS should be used only if the data are balanced, including no missing data.
• The order the terms appear in the model is important in the sum of square value calculated.
• The Sum of Square calculated is adjusted for the term(s) that appear before it in the
model statement.
• For example, if the model is yield=A B A*B, the Type I SS are calculated as o SS A o SS B|A (SS B given A is in the model o SS A*B|A, B (SS A*B given A and B are in the model
• THESE SUM SQUARES SHOULD NOT BE USED FOR HYPOTHESIS TESTING
UNLESS YOU HAVE BALANCED DATA, INCLUDING NO MISSING DATA. Type III SS
• THESE SUMS OF SQUARE ARE THE APPROPRIATE ONES TO USE FOR F-TESTS IF YOU HAVE UNBALANCED DATA, INCLUDING MISSING DATA.
• The order of terms in the model statement does not affect the calculated sums of squares.
• The Type III sums of squares are calculated assuming all other terms are in the model. If
you have a two-factor factorial, the sums of squares would be calculated as:
o SS A|B, AxB (SS A given B and AxB are in the model) o SS B|A, AxB (SS B given A and AxB are in the model) o SS AxB|A, B (SS A*B given A and B are in the model)
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
Class Level Information
Class Levels Values
Rep 4 1 2 3 4 A 2 0 1 B 2 0 1
Number of Observations Read 16 Number of Observations Used 16
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
Dependent Variable: Yield
Source DF Sum of
Squares Mean Square F Value Pr > F
Model 6 1149.000000 191.500000 82.07 <.0001 Error 9 21.000000 2.333333 Corrected Total 15 1170.000000
R-Square Coeff Var Root MSE Yield Mean
0.982051 6.110101 1.527525 25.00000
Source DF Type I SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A 1 930.2500000 930.2500000 398.68 <.0001 B 1 182.2500000 182.2500000 78.11 <.0001 A*B 1 4.0000000 4.0000000 1.71 0.2229
Source DF Type III SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A 1 930.2500000 930.2500000 398.68 <.0001 B 1 182.2500000 182.2500000 78.11 <.0001 A*B 1 4.0000000 4.0000000 1.71 0.2229
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
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30
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Yield
0 1
A
Distribution of Yield
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
t Tests (LSD) for Yield
Note: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05 Error Degrees of Freedom 9 Error Mean Square 2.333333 Critical Value of t 2.26216 Least Significant Difference 1.7278
1
0
32.6250
17.3750
A Estimate
Yield t Grouping for Means of A (Alpha = 0.05)Means covered by the same bar are not significantly different.
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
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Yield
0 1
B
Distribution of Yield
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
t Tests (LSD) for Yield
Note: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05 Error Degrees of Freedom 9 Error Mean Square 2.333333 Critical Value of t 2.26216 Least Significant Difference 1.7278
1
0
28.3750
21.6250
B Estimate
Yield t Grouping for Means of B (Alpha = 0.05)Means covered by the same bar are not significantly different.
Factorial, A and B fixed, and model order is A, B, and A*B
The GLM Procedure
Level of A
Level of B N
Yield
Mean Std Dev
0 0 4 13.5000000 1.29099445 0 1 4 21.2500000 1.70782513 1 0 4 29.7500000 2.50000000 1 1 4 35.5000000 2.64575131
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20
25
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35
Yie
ld
0 0 0 1 1 0 1 1
A*B
Distribution of Yield
Factorial, A and B fixed, and model order is A*B, B, and A
The GLM Procedure
Class Level Information
Class Levels Values
Rep 4 1 2 3 4 A 2 0 1 B 2 0 1
Number of Observations Read 16 Number of Observations Used 16
Factorial, A and B fixed, and model order is A*B, B, and A
The GLM Procedure
Dependent Variable: Yield
Source DF Sum of
Squares Mean Square F Value Pr > F
Model 6 1149.000000 191.500000 82.07 <.0001 Error 9 21.000000 2.333333 Corrected Total 15 1170.000000
R-Square Coeff Var Root MSE Yield Mean
0.982051 6.110101 1.527525 25.00000
Source DF Type I SS Mean Square F Value Pr > F
Rep 3 32.500000 10.833333 4.64 0.0317 A*B 3 1116.500000 372.166667 159.50 <.0001 B 0 0.000000 . . . A 0 0.000000 . . .
Source DF Type III SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A*B 1 4.0000000 4.0000000 1.71 0.2229 B 1 182.2500000 182.2500000 78.11 <.0001 A 1 930.2500000 930.2500000 398.68 <.0001
Factorial, A and B random, and model order is A, B, and A*B Output as it Would Appear on the Exam
The GLM Procedure
Class Level Information
Class Levels Values
Rep 4 1 2 3 4 A 2 0 1 B 2 0 1
Number of Observations Read 16 Number of Observations Used 16
Factorial, A and B random, and model order is A, B, and A*B Output as it Would Appear on the Exam
The GLM Procedure
Dependent Variable: Yield
Source DF Sum of
Squares Mean Square F Value Pr > F
Model 6 1149.000000 191.500000 82.07 <.0001 Error 9 21.000000 2.333333 Corrected Total 15 1170.000000
R-Square Coeff Var Root MSE Yield Mean
0.982051 6.110101 1.527525 25.00000
Source DF Type I SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A 1 930.2500000 930.2500000 398.68 <.0001 B 1 182.2500000 182.2500000 78.11 <.0001 A*B 1 4.0000000 4.0000000 1.71 0.2229
Source DF Type III SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A 1 930.2500000 930.2500000 398.68 <.0001 B 1 182.2500000 182.2500000 78.11 <.0001 A*B 1 4.0000000 4.0000000 1.71 0.2229
Factorial, A and B random, and model order is A, B, and A*B Output as it Would Appear on the Exam
The GLM Procedure
Level of A N
Yield
Mean Std Dev
0 8 17.3750000 4.37321392 1 8 32.6250000 3.88908730
Level of B N
Yield
Mean Std Dev
0 8 21.6250000 8.87914893 1 8 28.3750000 7.89099849
Level of A
Level of B N
Yield
Mean Std Dev
0 0 4 13.5000000 1.29099445 0 1 4 21.2500000 1.70782513 1 0 4 29.7500000 2.50000000 1 1 4 35.5000000 2.64575131
Factorial, A and B random, model order is a, B, and A*B SAS Code is for an all random model
YOU WILL NOT SEE THIS FORMAT ON PLSC 724 EXAMS
The GLM Procedure
Class Level Information
Class Levels Values
Rep 4 1 2 3 4 A 2 0 1 B 2 0 1
Number of Observations Read 16 Number of Observations Used 16
Factorial, A and B random, model order is a, B, and A*B SAS Code is for an all random model
YOU WILL NOT SEE THIS FORMAT ON PLSC 724 EXAMS
The GLM Procedure
Dependent Variable: Yield
Source DF Sum of
Squares Mean Square F Value Pr > F
Model 6 1149.000000 191.500000 82.07 <.0001 Error 9 21.000000 2.333333 Corrected Total 15 1170.000000
R-Square Coeff Var Root MSE Yield Mean
0.982051 6.110101 1.527525 25.00000
Source DF Type III SS Mean Square F Value Pr > F
Rep 3 32.5000000 10.8333333 4.64 0.0317 A 1 930.2500000 930.2500000 398.68 <.0001 B 1 182.2500000 182.2500000 78.11 <.0001 A*B 1 4.0000000 4.0000000 1.71 0.2229
Tests of Hypotheses Using the Type III MS for A*B as an Error Term
Source DF Type III SS Mean Square F Value Pr > F
A 1 930.2500000 930.2500000 232.56 0.0417 B 1 182.2500000 182.2500000 45.56 0.0936
Factorial, A and B random, model order is a, B, and A*B SAS Code is for an all random model
YOU WILL NOT SEE THIS FORMAT ON PLSC 724 EXAMS
The GLM Procedure
t Tests (LSD) for Yield
Note: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05 Error Degrees of Freedom 1 Error Mean Square 4 Critical Value of t 12.70620 Least Significant Difference 12.706
Means with the same letter are not significantly different.
t Grouping Mean N A
A 32.625 8 1
B 17.375 8 0
The GLM Procedure
YOU WILL NOT SEE THIS FORMAT ON PLSC 724 EXAMS
t Tests (LSD) for Yield
Note: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05 Error Degrees of Freedom 1 Error Mean Square 4 Critical Value of t 12.70620 Least Significant Difference 12.706
Means with the same letter are not significantly
different.
t Grouping Mean N B
A 28.375 8 1
A
A 21.625 8 0
Facotiral, A and B random, model order is a, B, and A*B SAS Code is for an all random model
YOU WILL NOT SEE THIS FORMAT ON PLSC 724 EXAMS
The GLM Procedure
Level of A
Level of B N
Yield
Mean Std Dev
0 0 4 13.5000000 1.29099445 0 1 4 21.2500000 1.70782513 1 0 4 29.7500000 2.50000000 1 1 4 35.5000000 2.64575131
Analysis Using PROC MIXED, A and B both random OPTIONAL: NOT REQUIRED TO KNOW FOR PLSC 724 EXAMS
The Mixed Procedure
Model Information
Data Set WORK.FACTORIAL Dependent Variable Yield Covariance Structure Variance Components Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment
Class Level Information
Class Levels Values
Rep 4 1 2 3 4 A 2 0 1 B 2 0 1
Dimensions
Covariance Parameters 5 Columns in X 1 Columns in Z 12 Subjects 1 Max Obs per Subject 16
Number of Observations
Number of Observations Read 16 Number of Observations Used 16 Number of Observations Not Used 0
Analysis Using PROC MIXED, A and B both random OPTIONAL: NOT REQUIRED TO KNOW FOR PLSC 724 EXAMS
The Mixed Procedure
Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 110.69137712 1 1 73.54143596 0.00000000
Convergence criteria met.
Covariance Parameter Estimates
Cov Parm Estimate
Rep 2.1250 A 115.78 B 22.2813 A*B 0.4167 Residual 2.3333
Fit Statistics
-2 Res Log Likelihood 73.5 AIC (Smaller is Better) 83.5 AICC (Smaller is Better) 90.2 BIC (Smaller is Better) 80.5
Solution for Random Effects
Effect Rep A B Estimate Std Err
Pred DF t Value Pr > |t|
Rep 1 -1.5692 0.9352 9 -1.68 0.1276 Rep 2 -0.1962 0.9352 9 -0.21 0.8385 Rep 3 1.5692 0.9352 9 1.68 0.1276 Rep 4 0.1962 0.9352 9 0.21 0.8385 A 0 -7.5922 7.6249 9 -1.00 0.3454 A 1 7.5922 7.6249 9 1.00 0.3454
Analysis Using PROC MIXED, A and B both random OPTIONAL: NOT REQUIRED TO KNOW FOR PLSC 724 EXAMS
The Mixed Procedure
Solution for Random Effects
Effect Rep A B Estimate Std Err
Pred DF t Value Pr > |t|
B 0 -3.3009 3.3742 9 -0.98 0.3535 B 1 3.3009 3.3742 9 0.98 0.3535 A*B 0 0 -0.2529 0.6100 9 -0.41 0.6882 A*B 0 1 0.2255 0.6100 9 0.37 0.7201 A*B 1 0 0.1911 0.6100 9 0.31 0.7612 A*B 1 1 -0.1638 0.6100 9 -0.27 0.7943
