SLIDE 1
Agenda
1
- 1. More on multilevel R formulas
- 2. Generalized Multilevel Models
- 3. GMLM in R
2
Agenda 1. More on multilevel R formulas 2. Generalized Multilevel - - PowerPoint PPT Presentation
Agenda 1. More on multilevel R formulas 2. Generalized Multilevel - - PowerPoint PPT Presentation
Agenda 1. More on multilevel R formulas 2. Generalized Multilevel Models 3. GMLM in R 1 More on multilevel R formulas 2 Building a two-level model <latexit
SLIDE 2
SLIDE 3
Building a two-level model
3
Our two-level model from last class
Sik ∼ Norm(μik, σ) μik = β0k + βkAgei + β2kFemalei + β3kBlacki β0k = γ00 + γ0Sizek + γ02Expi+ γ03TeacherFemalek+ γ04TeacherBlackk+ γ05PropBlackk + η0k βk = γ0 + γSizek + ηk β2k = γ20 + γ2TeacherFemalek + η2k β3k = γ30 + γ3TeacherBlackk+ γ32PropBlackk + η3k
<latexit sha1_base64="+Rpt2uKXCvMcGxWzJWs4kWPItZw=">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</latexit> SLIDE 4
Expanded notation
4
μik = β0k + βkAgei + β2kFemalei + β3kBlacki β0k = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + η0k βk = γ0 + γSizek + ηk β2k = γ20 + γ2TeacherFemalek + η2k β3k = γ30 + γ3TeacherBlackk + γ32PropBlackk + η3k
<latexit sha1_base64="z/O2lr1rLKTxBIZ91CfXy1f8Qyc=">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</latexit>μik = (γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + η0k)
(γ0 + γSizek + ηk)Agei (γ20 + γ2TeacherFemalek + η2k)Femalei (γ30 + γ3TeacherBlackk + γ32PropBlackk + η3k)Blacki
<latexit sha1_base64="4Ry1Auibz5N21MygTarOFdRWbKA=">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</latexit> SLIDE 5
Expanded notation
5
μik = (γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + η0k)
(γ0 + γSizek + ηk)Agei (γ20 + γ2TeacherFemalek + η2k)Femalei (γ30 + γ3TeacherBlackk + γ32PropBlackk + η3k)Blacki
Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">AG43icnVRLb9NAEDZtCRCehSOXFRWoKFlJ3AIVIpz0MphbRQKYmizXaSrLJrW7troCz7C7ghrvwzvwRxnYSnNLyWiXS+pvxN98PL1YcG18/9uZufmFs6Vz5y+UL16fOXqtcXr3WUKAb7LBKROuhRDYKHsG+4EXAQK6CyJ+BNb/Qwtb95C0rzKNwzRzF0JB2EvM8ZNQh1F+c/tmXStXzkyJ0GW4PqJS0a3fkQqZPgWuyT9Ad1TEau7x+7jLSaW8B5QNQT0BSQU6Vdrt8p12PKShiaS1rmGdm7616sbeW4KyUeo8say5XRXFUxhMCo7cXWQjx+l+Cg1mhAZFoRlDgAwPBtDlv6epzdDUAncspylfDfnG4B8o6zOU9eBY4kVb7YTU6xgoR9I4+GtOu/Q/TSK/JHRCl8i/tqlYet9N6nwqLbYn8l1VgrFn5R0tgkT9C8GjBTKPi5bpVD7MfS7FoxdZjpxwiBP5rI4YVlSxREpjPp5pg/g2LNBX59vHtyV8J/PpGbZ3gZX2tvpFeaqvB6rpPghU/O0ubxMvObndxbqd9GLFEQmiYoFq3Aj82HUuV4UyAK7cTDTEGogNo4TWkEnTHZnvCkduIHJ+pPAfGpKhxTcslVofyR56SmqG+rgtBU+0vc8DlNuH0MdFlD1ZedQTCTj76umWs/W16tp6deO+K89KNP17HcvDODEQslxhPxHERCTdVuSQK2BGHOGFMsUxScKGVFmcKeVCZ6ijExYDMyhJZVCE2HSTPtATaJA20+yG+NPTioNhk2qfGcGjbcjt6BYrhHq9nKbIy15xE0GEl5mDK1OvYZiLeAQijZgdQlN2Pg1L78iA+40dVtXK9h9akCGN1t7Syh3IbGD7k4aCajlik0v1LhZsJeDq7kq0/ODbBNLF3kXiCOqygagA6e9bOhpHCgXT2xfMDZ/k7DOysRNiVcQInY0ZOv7yurQT1leDl6tLmVj6K3nvpnfLW/YCb8Pb9J5u96+x+a/L1xYuL5wowSlT6XPpS+569yZ8Ts3vJlT+voDGkxcSA=</latexit> SLIDE 6
Expanded notation
6
Random effects
Unexplained variation in outcome variable. Described in terms of individual variability and different types of group variability.
Fixed effects
Explained variation in
- utcome variable.
Describes the way that
- utcome and predictor
variables co-vary.
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit> SLIDE 7
Building an R formula
7
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>With level-two covariates interactions need to be specified
SLIDE 8
Building an R formula
8
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Outcome variable
SLIDE 9
Building an R formula
9
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Global intercept (included automatically)
SLIDE 10
Building an R formula
10
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Interactions (*) automatically include standalone terms
SLIDE 11
Building an R formula
11
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">AG43icnVRLb9NAEDZtCRCehSOXFRWoKFlJ3AIVIpz0MphbRQKYmizXaSrLJrW7troCz7C7ghrvwzvwRxnYSnNLyWiXS+pvxN98PL1YcG18/9uZufmFs6Vz5y+UL16fOXqtcXr3WUKAb7LBKROuhRDYKHsG+4EXAQK6CyJ+BNb/Qwtb95C0rzKNwzRzF0JB2EvM8ZNQh1F+c/tmXStXzkyJ0GW4PqJS0a3fkQqZPgWuyT9Ad1TEau7x+7jLSaW8B5QNQT0BSQU6Vdrt8p12PKShiaS1rmGdm7616sbeW4KyUeo8say5XRXFUxhMCo7cXWQjx+l+Cg1mhAZFoRlDgAwPBtDlv6epzdDUAncspylfDfnG4B8o6zOU9eBY4kVb7YTU6xgoR9I4+GtOu/Q/TSK/JHRCl8i/tqlYet9N6nwqLbYn8l1VgrFn5R0tgkT9C8GjBTKPi5bpVD7MfS7FoxdZjpxwiBP5rI4YVlSxREpjPp5pg/g2LNBX59vHtyV8J/PpGbZ3gZX2tvpFeaqvB6rpPghU/O0ubxMvObndxbqd9GLFEQmiYoFq3Aj82HUuV4UyAK7cTDTEGogNo4TWkEnTHZnvCkduIHJ+pPAfGpKhxTcslVofyR56SmqG+rgtBU+0vc8DlNuH0MdFlD1ZedQTCTj76umWs/W16tp6deO+K89KNP17HcvDODEQslxhPxHERCTdVuSQK2BGHOGFMsUxScKGVFmcKeVCZ6ijExYDMyhJZVCE2HSTPtATaJA20+yG+NPTioNhk2qfGcGjbcjt6BYrhHq9nKbIy15xE0GEl5mDK1OvYZiLeAQijZgdQlN2Pg1L78iA+40dVtXK9h9akCGN1t7Syh3IbGD7k4aCajlik0v1LhZsJeDq7kq0/ODbBNLF3kXiCOqygagA6e9bOhpHCgXT2xfMDZ/k7DOysRNiVcQInY0ZOv7yurQT1leDl6tLmVj6K3nvpnfLW/YCb8Pb9J5u96+x+a/L1xYuL5wowSlT6XPpS+569yZ8Ts3vJlT+voDGkxcSA=</latexit>Interactions (*) automatically include standalone terms Interactions (*) automatically include standalone terms
SLIDE 12
Building an R formula
12
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">AG43icnVRLb9NAEDZtCRCehSOXFRWoKFlJ3AIVIpz0MphbRQKYmizXaSrLJrW7troCz7C7ghrvwzvwRxnYSnNLyWiXS+pvxN98PL1YcG18/9uZufmFs6Vz5y+UL16fOXqtcXr3WUKAb7LBKROuhRDYKHsG+4EXAQK6CyJ+BNb/Qwtb95C0rzKNwzRzF0JB2EvM8ZNQh1F+c/tmXStXzkyJ0GW4PqJS0a3fkQqZPgWuyT9Ad1TEau7x+7jLSaW8B5QNQT0BSQU6Vdrt8p12PKShiaS1rmGdm7616sbeW4KyUeo8say5XRXFUxhMCo7cXWQjx+l+Cg1mhAZFoRlDgAwPBtDlv6epzdDUAncspylfDfnG4B8o6zOU9eBY4kVb7YTU6xgoR9I4+GtOu/Q/TSK/JHRCl8i/tqlYet9N6nwqLbYn8l1VgrFn5R0tgkT9C8GjBTKPi5bpVD7MfS7FoxdZjpxwiBP5rI4YVlSxREpjPp5pg/g2LNBX59vHtyV8J/PpGbZ3gZX2tvpFeaqvB6rpPghU/O0ubxMvObndxbqd9GLFEQmiYoFq3Aj82HUuV4UyAK7cTDTEGogNo4TWkEnTHZnvCkduIHJ+pPAfGpKhxTcslVofyR56SmqG+rgtBU+0vc8DlNuH0MdFlD1ZedQTCTj76umWs/W16tp6deO+K89KNP17HcvDODEQslxhPxHERCTdVuSQK2BGHOGFMsUxScKGVFmcKeVCZ6ijExYDMyhJZVCE2HSTPtATaJA20+yG+NPTioNhk2qfGcGjbcjt6BYrhHq9nKbIy15xE0GEl5mDK1OvYZiLeAQijZgdQlN2Pg1L78iA+40dVtXK9h9akCGN1t7Syh3IbGD7k4aCajlik0v1LhZsJeDq7kq0/ODbBNLF3kXiCOqygagA6e9bOhpHCgXT2xfMDZ/k7DOysRNiVcQInY0ZOv7yurQT1leDl6tLmVj6K3nvpnfLW/YCb8Pb9J5u96+x+a/L1xYuL5wowSlT6XPpS+569yZ8Ts3vJlT+voDGkxcSA=</latexit>Interactions (*) automatically include standalone terms Interactions (*) automatically include standalone terms
SLIDE 13
Building an R formula
13
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Redundant terms (student_re_black) are not added twice
SLIDE 14
Building an R formula
14
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Random effects use pipe notation (|)
SLIDE 15
Building an R formula
15
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">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</latexit>Grouping elements after the pipe
SLIDE 16
Building an R formula
16
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
<latexit sha1_base64="7TN0SMGmd2lFRM3eY1JhqgaJRaE=">AG43icnVRLb9NAEDZtCRCehSOXFRWoKFlJ3AIVIpz0MphbRQKYmizXaSrLJrW7troCz7C7ghrvwzvwRxnYSnNLyWiXS+pvxN98PL1YcG18/9uZufmFs6Vz5y+UL16fOXqtcXr3WUKAb7LBKROuhRDYKHsG+4EXAQK6CyJ+BNb/Qwtb95C0rzKNwzRzF0JB2EvM8ZNQh1F+c/tmXStXzkyJ0GW4PqJS0a3fkQqZPgWuyT9Ad1TEau7x+7jLSaW8B5QNQT0BSQU6Vdrt8p12PKShiaS1rmGdm7616sbeW4KyUeo8say5XRXFUxhMCo7cXWQjx+l+Cg1mhAZFoRlDgAwPBtDlv6epzdDUAncspylfDfnG4B8o6zOU9eBY4kVb7YTU6xgoR9I4+GtOu/Q/TSK/JHRCl8i/tqlYet9N6nwqLbYn8l1VgrFn5R0tgkT9C8GjBTKPi5bpVD7MfS7FoxdZjpxwiBP5rI4YVlSxREpjPp5pg/g2LNBX59vHtyV8J/PpGbZ3gZX2tvpFeaqvB6rpPghU/O0ubxMvObndxbqd9GLFEQmiYoFq3Aj82HUuV4UyAK7cTDTEGogNo4TWkEnTHZnvCkduIHJ+pPAfGpKhxTcslVofyR56SmqG+rgtBU+0vc8DlNuH0MdFlD1ZedQTCTj76umWs/W16tp6deO+K89KNP17HcvDODEQslxhPxHERCTdVuSQK2BGHOGFMsUxScKGVFmcKeVCZ6ijExYDMyhJZVCE2HSTPtATaJA20+yG+NPTioNhk2qfGcGjbcjt6BYrhHq9nKbIy15xE0GEl5mDK1OvYZiLeAQijZgdQlN2Pg1L78iA+40dVtXK9h9akCGN1t7Syh3IbGD7k4aCajlik0v1LhZsJeDq7kq0/ODbBNLF3kXiCOqygagA6e9bOhpHCgXT2xfMDZ/k7DOysRNiVcQInY0ZOv7yurQT1leDl6tLmVj6K3nvpnfLW/YCb8Pb9J5u96+x+a/L1xYuL5wowSlT6XPpS+569yZ8Ts3vJlT+voDGkxcSA=</latexit>Random intercepts indicated with constant (1)
SLIDE 17
Building an R formula
17
student_reading_score ~ student_age_s*class_size_c + teacher_exper_c + student_female*teacher_female + student_re_black*teacher_re_black + student_re_black*class_prop_black + (1 + student_age_s + student_female + student_re_black | teacher_id)
μik γ00 γ0Sizek γ02Expi TeacherFemalek γ04TeacherBlackk γ05PropBlackk η0k γ0 γSizek ηk Agei γ20 γ2TeacherFemalek η2k Femalei γ30 γ3TeacherBlackk γ32PropBlackk η3k Blacki Sik = γ00 + γ0Sizek + γ02Expi + TeacherFemalek+ γ04TeacherBlackk + γ05PropBlackk + γ0Agei γAgeiSizek + γ20Femalei + γ2FemaleiTeacherFemalek+ γ30Blacki + γ3BlackiTeacherBlackk + γ32BlackiPropBlackk+ η0k + ηkAgei + η2kFemalei + η3kBlacki + εi
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SLIDE 18
18
Generalized multilevel models
SLIDE 19
Generalized multilevel models
19
Generalized multilevel linear models
Simply add a link function and change the outcome distribution. E.g. modelling whether a student did better on the math test than the reading test (Mik).
Mik ∼ Binomial(, pik) logit(pik) = β0k + βkAgei + β2kFemalei β0k = γ00 + γ0TExpk + γ02TFemalek + η0k βk = γ0 + ηk β2k = γ20 + γ2TFemalek + η2k
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Generalized multilevel models
20
This is where the ‘generalized’ part
- f the model lives
Mik ∼ Binomial(, pik) logit(pik) = β0k + βkAgei + β2kFemalei β0k = γ00 + γ0TExpk + γ02TFemalek + η0k βk = γ0 + ηk β2k = γ20 + γ2TFemalek + η2k
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Generalized multilevel models
21
This is the same as a standard (Gaussian) multilevel model Mik ∼ Binomial(, pik) logit(pik) = β0k + βkAgei + β2kFemalei β0k = γ00 + γ0TExpk + γ02TFemalek + η0k βk = γ0 + ηk β2k = γ20 + γ2TFemalek + η2k
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Generalized multilevel models
22
Generalized multilevel linear models
Simply add a link function and change the outcome distribution. E.g. modelling whether a student did better on the reading test than the math test (Mik).
All coefficients and parameters are affected by link function
Interpretation requires careful thinking about the ways that coefficients affect pik. Interpreting direction of effect (positive vs. negative) is still straightforward. E.g. a strong positive estimate on γ02 would suggest that female teachers do a relatively better job of teaching mathematics.
Mik ∼ Binomial(, pik) logit(pik) = β0k + βkAgei + β2kFemalei β0k = γ00 + γ0TExpk + γ02TFemalek + η0k βk = γ0 + ηk β2k = γ20 + γ2TFemalek + η2k
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