Long-Term Correlates of Family Foster Care

Report
Generalized Linear Model (GZLM):
Overview
Dependent Variables
Continuous
 Discrete

 Dichotomous
 Polychotomous
 Ordinal
 Count
Continuous Variables

Quantitative variables that can take on any
value within the limits of the variable
Continuous Variables (cont’d)

Distance, time, or length
 Infinite number of possible divisions between
any two values, at least theoretically
 “Only love can be divided endlessly and still not
diminish” (Anne Morrow Lindbergh)

More than 11 ordered values
 Scores on standardized scales such as those
that measure parenting attitudes, depression,
family functioning, and children’s behavioral
problems
Discrete Variables

Finite number of indivisible values; cannot
take on all possible values within the limits
of the variable
 Dichotomous
 Polytomous
 Ordinal
 Count
Dichotomous Variables
Two categories used to indicate whether
an event has occurred or some
characteristic is present
 Sometimes called binary or binomial
variables
 “To be or not to be, that is the question..”
(William Shakespeare, “Hamlet”)

Dichotomous DVs





Placed in foster care or not
Diagnosed with a disease or not
Abused or not
Pregnant or not
Service provided or not
Polytomous Variables
Three or more unordered categories
 Categories mutually exclusive and
exhaustive
 Sometimes called multicategorical or
sometimes multinomial variables
 “Inanimate objects can be classified
scientifically into three major categories;
those that don't work, those that break
down and those that get lost” (Russell
Baker)

Polytomous DVs



Reason for leaving welfare:
 marriage, stable employment, move to another
state, incarceration, or death
Status of foster home application:
 licensed to foster, discontinued application
process prior to licensure, or rejected for
licensure
Changes in living arrangements of the elderly:
 newly co-residing with their children, no longer
co-residing, or residing in institutions
Ordinal Variables
Three or more ordered categories
 Sometimes called ordered categorical
variables or ordered polytomous variables
 “Good, better, best; never let it rest till
your good is better and your better is
best” (Anonymous)

Ordinal DVs

Job satisfaction:
 very dissatisfied, somewhat dissatisfied,
neutral, somewhat satisfied, or very satisfied

Severity of child abuse injury:
 none, mild, moderate, or severe

Willingness to foster children with
emotional or behavioral problems:
 least acceptable, willing to discuss, or most
acceptable
Count Variables

Number of times a particular event occurs
to each case, usually within a given:
 Time period (e.g., number of hospital visits per
year)
 Population size (e.g., number of registered sex
offenders per 100,000 population), or
 Geographical area (e.g., number of divorces per
county or state)

Whole numbers that can range from 0
through +
Count Variables (cont’d)
“Now I've got heartaches by the number,
Troubles by the score,
Every day you love me less,
Each day I love you more” (Ray Price)
Count DVs

Number of hospital visits, outpatient visits,
services used, divorces, arrests, criminal
offenses, symptoms, placements, children
fostered, children adopted
General Linear Model (GLM)
(selected models)
Continuous DV
Linear Regression
ANOVA
t-test
Generalized Linear Model (GZLM)
(selected regression models)
GZLM
Continuous
DV
Dichotomous
DV
Polytomous
DV
Ordinal
DV
Count
DV
Linear
Regression
Binary
Logistic
Regression
Multinomial
Logistic
Regression
Ordinal
Logistic
Regression
Poisson or
Negative
Binomial
Regression
Generalized How?
DV continuous or discrete
 Normal or non-normal error distributions
 Constant or non-constant variance
 Provides a unifying framework for
analyzing an entire class of regression
models

GLM & GZLM Similarities
IVs are combined in a linear fashion (α +
1X1 + 2X2 + … kXk ;
 a slope is estimated for each IV;
 each slope has an accompanying test of
statistical significance and confidence
interval;
 each slope indicates the IV’s independent
contribution to the explanation or
prediction of the DV;

GLM & GZLM Similarities (cont’d)





the sign of each slope indicates the
direction of the relationship
IVs can be any level of measurement;
the same methods are used for coding
categorical IVs (e.g., dummy coding);
IVs can be entered simultaneously,
sequentially or using other methods;
product terms can be used to test
interactions;
GLM & GZLM Similarities (cont’d)
powered terms (e.g., the square of an IV)
can be used to test curvilinearity;
 overall model fit can be tested, as can
incremental improvement in a model
brought about by the addition or deletion
of IVs (nested models); and
 residuals, leverage values, Cook’s D, and
other indices are used to diagnose model
problems.

Common Assumptions
Correct model specification
 Variables measured without error
 Independent errors
 No perfect multicollinearity

Correct Model Specification
Have you included relevant IVs?
 Have you excluded irrelevant IVs?
 Do the IVs that you have included have
linear or non-linear relationships with your
DV (or some function of your DV, as
discussed below)?
 Are one or more of your IVs moderated by
other IVs (i.e., are there interaction
effects)?

Variables Measured without Error

Limitation of regression models, given that
most often our variables contain some
measurement error
Independent Errors

Can be result of study design, e.g.:
– Clustered data, which occurs when data are
collected from groups
– Temporally linked data, which occurs when data
are collected repeatedly over time from the
same people or groups

Can lead to incorrect significance tests and
confidence intervals
Independent Errors (cont’d)

Examples of when this might not be true
 Effect of parenting practices on behavioral
problems of children and reports of parenting
practices and behavioral problems collected
from both parents in two-parent families
 Effect of parenting practices on behavioral
problems of children and information collected
about behavioral problems for two or more
children per family
 Effects of leader behaviors on group cohesion
in small groups, and information collected about
leader behaviors and group cohesion from all
members of each group
No Perfect Multicollinearity
Perfect multicollinearity exists when an IV
is predicted perfectly by a linear
combination of the remaining IVs
 Typically quantified by “tolerance” or
“variance inflation factor” (VIF)
(1/tolerance)
 Even high levels of multicollinearity may
pose problems (e.g., tolerance < .20 or
especially < .10)

Estimating Parameters (e.g.,

)
GLM
 Ordinary Least Squares (OLS) estimation
• Estimates minimize sum of the squared differences
between observed and estimated values of the DV
http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html

GZLM
 Maximum Likelihood (ML) estimation
• Estimates have greatest likelihood (i.e., the maximum
likelihood) of generating observed sample data if
model assumptions are true
Testing Hypotheses

Overall and nested models (1 = 2 = k = 0)
 GLM
• F
 GZLM
• Likelihood ratio 2

Individual slopes ( = 0)
 GLM
• t
 GZLM
• Wald 2 or likelihood ratio 2
Estimating DV with GLM

Three ways of expressing the same thing…
  = α + 1X1 + 2X2 + … kXk
 = 
• Assumed linear relationship

 = Greek letter mu
 Estimated mean value of DV

 = Greek letter eta
 Linear predictor
Estimating DV with Poisson
Regresion
ln() = α + 1X1 + 2X2 + … kXk
 ln() = 

 Assumed linear relationship
Single (Quantitative) IV Example
DV = number of foster children adopted
 IV = Perceived responsibility for parenting
(scale scores transformed to z-scores)
 N = 285 foster mothers


Do foster mothers who feel a greater
responsibility to parent foster children
adopt more foster children?
Poisson Model

ln() = α + X

log of estimated mean count
 .018 + (.185)(X)
 Log of mean number of children adopted
 Does not have intuitive or substantive meaning
Mathematical Functions

Function
 √4 = 2

Inverse (reverse) function
 22 = 4
Mathematical Functions (cont’d)

Function
 ln(), natural logarithm of 
 “Link function”

Inverse (reverse) function
 exp(), exponential of 
• ex on calculator
• exp(x) in SPSS and Excel
 “Inverse link function”
Link Function

ln(), log of estimated mean count
 Connects (i.e., links) mean value of DV to linear
combination of IVs
 Transforms relationship between  and  so
relationship is linear
 Different GZLM models use different links
 Does not have intuitive or substantive meaning
Inverse (Reverse) Link Function

Three ways of expressing the same thing…
  = exp(α + 1X1 + 2X2 + … kXk)
  = exp()
  = e

 represent values of the DV with intuitive
and substantive meaning
 e.g., mean number of children adopted
Estimated Mean DV
.018 + (.185)(X)

X=0

X=1
 .018 + (.185)(0) = .018
 e.018 = 1.018
 M = 1.02 children adopted
 .018 + (.185)(1) = .203
 e.203 = 1.225
 M = 1.23 children adopted
Examples of Exponentiation

e0 = 1.00

e.50 = 1.65

e1.00 = 2.72
Problem

For discrete DVs the relationship between
the DV () and the linear predictor () is
non-linear
  = α + 1X1 + 2X2 + … kXk
 = 
• Non-linear
 One-unit increase in an IV may be associated
with a different amount of change in the mean
DV, depending on the initial value of the IV
Mean Number of Children
Example Non-linear Relationship
2.00
1.50
1.00
0.50
0.00
Mean Number of
Children
-3
-2
-1
0
1
2
3
0.58
0.70
0.85
1.02
1.23
1.47
1.77
Standardized Parenting Responsibility
Solution

Linear relationship between a linear
combination of one or more IVs and some
function of the DV
ln(Mean Number of Children)
Example Linear Relationship
0.80
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
ln(Mean Number of
Children)
-3
-2
-1
0
1
2
3
-0.54
-0.35
-0.17
0.02
0.20
0.39
0.57
Standardized Parenting Responsibility

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