SPSS混合线性模型
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Leabharlann Baidu
grand mean (an unknown fixed parm) i effect of ith value of (ai )
b j effect of jth value of b (b j )
ij exp erimental error ~ N (0, 2 )
Within-subject effects are experienced by subjects repeatedly over time. Trial is a random effect when there are several trials in the repeated measures design; all subjects experience all of the trials. Trial is therefore a within-subject effect. Operator may be a fixed or random effect, depending upon whether one is generalizing beyond the sample If operator is a random effect, then the machine*operator interaction is a random effect. There are contrasts: These contrast the values of one level with those of other levels of the same effect.
Level 1
X11
X12
X13
X14
Variable X Level 2 X21 X22 X23 X24
Level 3
X31
X32
X33
X34
7
Classification of Effectscont’d
Hierarchical designs have nested effects. Nested effects are those with subjects within groups. An example would be patients nested within doctors and doctors nested within hospitals This could be expressed by patients(doctors) doctors(hospitals)
13
The General Linear Model
1. The main effects general linear model can be parameterized as
Yij i b j ij where
Yij observation for ith
12
Repeated Observations are Within-Subjects effects
Repeated Measures Design
Trial 1
Experimental Group Pre-test
Trial 2
Experimental Group Post-test
Trial 3
1. The one-way layout
4. Mixed Model theory
1. Proper error terms
5. Two-way layout 6. Full-factorial model
1. Contrasts with interaction terms 2. Graphing Interactions
i j ( yij ) ( )
yij
15
Higher-Order Interactions
If 3-way interactions are in the model, then the main effects and all lower order interactions must be in the model for the 3way interaction to be properly specified. For example, a 3-way interaction model would be:
Mixed Analysis of Variance Models with SPSS
Robert A.Yaffee, Ph.D. Statistics, Social Science, and Mapping Group Information Technology Services/Academic Computing Services Office location: 75 Third Avenue, Level C-3 Phone: 212-998-3402
10
Between Subject effects
• Gender: One is either male or female, but not both. • Group: One is either in the control, experimental, or the comparison group but not more than one.
14
A factorial model
If an interaction term were included, the formula would be
yij i i ij eij
The interaction or crossed effect is the joint effect, over and above the individual main effects. Therefore, the main effects must be in the model for the interaction to be properly specified.
6
Interactions are Crossed Effects
All of the cells are filled Each level of X is crossed with each level of Y Variable Y Level 1 Level 2 Level 3 Level 4
1. Subject: the sample is a random sample of the target population
5
Classification of effects
1. There are main effects: Linear Explanatory Factors 2. There are interaction effects: Joint effects over and above the component main effects.
Experimental Group Follow-up
Group
Control Group Pre-test Control Group Post-test Control Group Follow-up
Group is a between subjects effect, whereas Trial is a within subjects effect.
11
Within-Subjects Effects
• These are repeated effects. • Observation 1, 2, and 3 might be the pre, post, and follow-up observations on each person. • Each person experiences all of these levels or categories. • These are found in repeated measures analysis of variance.
Pat 7
Pat 8
9
Between and WithinSubject effects
• Such effects may sometimes be fixed or random. Their classification depends on the experimental design Between-subjects effects are those who are in one group or another but not in both. Experimental group is a fixed effect because the manager is considering only those groups in his experiment. One group is the experimental group and the other is the control group. Therefore, this grouping factor is a between- subject effect.
1
Outline
1. Classification of Effects 2. Random Effects
1. Two-Way Random Layout 2. Solutions and estimates
3. General linear model
1. Fixed Effects Models
4
Examples of Fixed and Random Effects
1. Fixed effect: 2. Sex where both male and female genders are included in the factor, sex. 3. Agegroup: Minor and Adult are both included in the factor of agegroup 4. Random effect:
2
Outline-Cont’d
• Repeated Measures ANOVA • Advantages of Mixed Models over GLM.
3
Definition of Mixed Models by their component effects
1. Mixed Models contain both fixed and random effects 2. Fixed Effects: factors for which the only levels under consideration are contained in the coding of those effects 3. Random Effects: Factors for which the levels contained in the coding of those factors are a random sample of the total number of levels in the population for that factor.
8
Nesting of patients within Doctors and Doctors within Hospitals
Hospital 1
Hospital 2
Doctor1
Doctor 2
Doctor 3
Doctor 4
Doctor 5
Pat 1
Pat 2
Pat 3
Pat 4
Pat 5
Pat 6
yijk ai b j ck abij acik bc jk abcijk eijk
16
The General Linear Model
• In matrix terminology, the general linear model may be expressed as
grand mean (an unknown fixed parm) i effect of ith value of (ai )
b j effect of jth value of b (b j )
ij exp erimental error ~ N (0, 2 )
Within-subject effects are experienced by subjects repeatedly over time. Trial is a random effect when there are several trials in the repeated measures design; all subjects experience all of the trials. Trial is therefore a within-subject effect. Operator may be a fixed or random effect, depending upon whether one is generalizing beyond the sample If operator is a random effect, then the machine*operator interaction is a random effect. There are contrasts: These contrast the values of one level with those of other levels of the same effect.
Level 1
X11
X12
X13
X14
Variable X Level 2 X21 X22 X23 X24
Level 3
X31
X32
X33
X34
7
Classification of Effectscont’d
Hierarchical designs have nested effects. Nested effects are those with subjects within groups. An example would be patients nested within doctors and doctors nested within hospitals This could be expressed by patients(doctors) doctors(hospitals)
13
The General Linear Model
1. The main effects general linear model can be parameterized as
Yij i b j ij where
Yij observation for ith
12
Repeated Observations are Within-Subjects effects
Repeated Measures Design
Trial 1
Experimental Group Pre-test
Trial 2
Experimental Group Post-test
Trial 3
1. The one-way layout
4. Mixed Model theory
1. Proper error terms
5. Two-way layout 6. Full-factorial model
1. Contrasts with interaction terms 2. Graphing Interactions
i j ( yij ) ( )
yij
15
Higher-Order Interactions
If 3-way interactions are in the model, then the main effects and all lower order interactions must be in the model for the 3way interaction to be properly specified. For example, a 3-way interaction model would be:
Mixed Analysis of Variance Models with SPSS
Robert A.Yaffee, Ph.D. Statistics, Social Science, and Mapping Group Information Technology Services/Academic Computing Services Office location: 75 Third Avenue, Level C-3 Phone: 212-998-3402
10
Between Subject effects
• Gender: One is either male or female, but not both. • Group: One is either in the control, experimental, or the comparison group but not more than one.
14
A factorial model
If an interaction term were included, the formula would be
yij i i ij eij
The interaction or crossed effect is the joint effect, over and above the individual main effects. Therefore, the main effects must be in the model for the interaction to be properly specified.
6
Interactions are Crossed Effects
All of the cells are filled Each level of X is crossed with each level of Y Variable Y Level 1 Level 2 Level 3 Level 4
1. Subject: the sample is a random sample of the target population
5
Classification of effects
1. There are main effects: Linear Explanatory Factors 2. There are interaction effects: Joint effects over and above the component main effects.
Experimental Group Follow-up
Group
Control Group Pre-test Control Group Post-test Control Group Follow-up
Group is a between subjects effect, whereas Trial is a within subjects effect.
11
Within-Subjects Effects
• These are repeated effects. • Observation 1, 2, and 3 might be the pre, post, and follow-up observations on each person. • Each person experiences all of these levels or categories. • These are found in repeated measures analysis of variance.
Pat 7
Pat 8
9
Between and WithinSubject effects
• Such effects may sometimes be fixed or random. Their classification depends on the experimental design Between-subjects effects are those who are in one group or another but not in both. Experimental group is a fixed effect because the manager is considering only those groups in his experiment. One group is the experimental group and the other is the control group. Therefore, this grouping factor is a between- subject effect.
1
Outline
1. Classification of Effects 2. Random Effects
1. Two-Way Random Layout 2. Solutions and estimates
3. General linear model
1. Fixed Effects Models
4
Examples of Fixed and Random Effects
1. Fixed effect: 2. Sex where both male and female genders are included in the factor, sex. 3. Agegroup: Minor and Adult are both included in the factor of agegroup 4. Random effect:
2
Outline-Cont’d
• Repeated Measures ANOVA • Advantages of Mixed Models over GLM.
3
Definition of Mixed Models by their component effects
1. Mixed Models contain both fixed and random effects 2. Fixed Effects: factors for which the only levels under consideration are contained in the coding of those effects 3. Random Effects: Factors for which the levels contained in the coding of those factors are a random sample of the total number of levels in the population for that factor.
8
Nesting of patients within Doctors and Doctors within Hospitals
Hospital 1
Hospital 2
Doctor1
Doctor 2
Doctor 3
Doctor 4
Doctor 5
Pat 1
Pat 2
Pat 3
Pat 4
Pat 5
Pat 6
yijk ai b j ck abij acik bc jk abcijk eijk
16
The General Linear Model
• In matrix terminology, the general linear model may be expressed as