Transcript Long-Term Correlates of Family Foster Care

Ordinal Logistic Regression
“Good, better, best; never let it rest till
your good is better and your better is best”
(Anonymous)
Ordinal Logistic Regression

Also known as the “ordinal logit,” “ordered
polytomous logit,” “constrained cumulative
logit,” “proportional odds,” “parallel
regression,” or “grouped continuous model”

Generalization of binary logistic regression
to an ordinal DV
 When applied to a dichotomous DV identical to
binary logistic regression
Ordinal Variables
Three or more ordered categories
 Sometimes called “ordered categorical” or
“ordered polytomous” variables

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
Single (Dichotomous) IV Example

DV = satisfaction with foster care agencies
 (1) dissatisfied; (2) neither satisfied nor
dissatisfied; (3) satisfied

IV = agencies provided sufficient
information about the role of foster care
workers
 0 (no) or 1 (yes)

N = 300 foster mothers
Single (Dichotomous) IV Example
(cont’d)

Are foster mothers who report that they
were provided sufficient information about
the role of foster care workers more
satisfied with their foster care agencies?
Crosstabulation

Table 4.1

Relationship between information and
satisfaction is statistically significant
[2(2, N = 300) = 23.52, p < .001]
Cumulative Probability

Ordinal logistic regression focuses on
cumulative probabilities of the DV and
odds and ORs based on cumulative
probabilities.
 By cumulative probability we mean the
probability that the DV is less than or equal to
a particular value (e.g., 1, 2, or 3 in our
example).
Cumulative Probabilities

Dissatisfied
 Insufficient Info: .2857
 Sufficient Info: .1151

Dissatisfied or neutral
 Insufficient Info: .5590 (.2857 + .2733)
 Sufficient Info: .2878 (.1151 + .1727)

Dissatisfied, neutral, or satisfied
 Insufficient Info: 1.00 (.2867 + .2733 + .4410)
 Sufficient Info: 1.00 (.1151 + .1727 + .7121)
Cumulative Odds

Probability that the DV is less than or
equal to a particular value is compared to
(divided by) the probability that it is
greater than that value
 Reverse of what you do in binary and
multinomial logistic regression
 Probability that the DV is 1 (dissatisfied) vs.
the probability that it is either 2 or 3 (neutral
or satisfied); probability that the DV is 1 or 2
(dissatisfied or neutral) vs. the probability that
it is 3 (satisfied)
Cumulative Odds & Odds Ratios

Odds of being dissatisfied (vs. neutral or
satisfied)
 Insufficient Info: .4000 (.2857 / [1 - .2857])
 Sufficient Info: .1301 (.1151 / [1 - .1151])
 OR = .33 (.1301 / .4000) (-67%)

Odds of being dissatisfied or neutral (vs.
satisfied)
 Insufficient Info: 1.2676 (.5590 / [1 - .5590])
 Sufficient Info: .4041 (.2878 / [1 - .2878])
 OR = .32 (.4041 / 1.2676) (-68%)
Question & Answer

Are foster mothers who report that they were
provided sufficient information about the role of
foster care workers more satisfied with their
foster care agencies?

The odds of being dissatisfied (vs. being neutral
or satisfied) are .33 times (67%) smaller for
mothers who received sufficient information. The
odds of being dissatisfied or neutral (vs. being
satisfied) are .32 times (68%) smaller for mothers
who received sufficient information.
Ordinal Logistic Regression

Set of binary logistic regression models
estimated simultaneously (like multinomial
logistic regression)
 Number of non-redundant binary logistic
regression equations equals the number of
categories of the DV minus one

Focus on cumulative probabilities and odds,
and ORs are computed from cumulative
odds (unlike multinomial logistic regression)
Threshold


Suppose our three-point variable is a rough
measure of an underlying continuous satisfaction
variable. At a certain point on this continuous
variable the population threshold (symbolized by τ,
the Greek letter tau), that is a person’s level of
satisfaction, goes from one value to another on
the ordinal measure of satisfaction.
e.g., the first threshold (τ1) would be the point at
which the level of satisfaction goes from
dissatisfied to neutral (i.e., 1 to 2), and the second
threshold (τ2) would be the point at which the
level of satisfaction goes from neutral to satisfied
(i.e., 2 to 3).
Threshold (cont’d)



The number of thresholds is always one fewer
than the number of values of the DV.
Usually thresholds are of little interest except in
the calculation of estimated values.
Thresholds typically are used in place of the
intercept to express the ordinal logistic
regression model
Estimated Cumulative Logits
L (Dissatisfied vs. Neutral/Satisfied) = t1 - BX
L (Dissatisfied/Neutral vs. Satisfied) = t2 – BX
Table 4.2
L (Dissatisfied vs. Neutral/Satisfied) = -.912 – 1.139X
L (Dissatisfied/Neutral vs. Satisfied) = .235 – 1.139X
Estimated Cumulative Logits
(cont’d)
Each equation has a different threshold
(e.g., t1 and t2)
 One common slope (B).

 It is assumed that the effect of the IVs is the
same for different values of the DV (“parallel
regression” assumption)

Slope is multiplied by a value of the IV and
subtracted from, not added to, the
threshold.
Statistical Significance

Table 4.2
(Info) = 0
• Reject
Estimated Cumulative Logits (X =
1)
L (Dissatisfied vs. Neutral/Satisfied) = -2.051 = -.912 – (1.139)(1)
L (Dissatisfied/Neutral vs. Satisfied) = -.904 = .235 – (1.139)(1)
Effect of Information on
Satisfaction (Cumulative Logits)
1.00
Logits
0.00
-1.00
-2.00
-3.00
(0) Insufficient
(1) Sufficient
Dissatisfied
-0.91
-2.05
Dissatisfied/
Neutral
0.23
-0.90
Information
Cumulative Logits to Cumulative
Odds (X = 1)
L (Dissatisfied vs. Neutral/Satisfied) = e-2.051 = .129
L (Dissatisfied/Neutral vs. Satisfied) = e-.904 = .405
Effect of Information on
Satisfaction (Cumulative Odds)
Odds
1.50
1.00
0.50
0.00
(0) Insufficient
(1) Sufficient
Dissatisfied
0.40
0.13
Dissatisfied/
Neutral
1.26
0.40
Information
Cumulative Logits to Cumulative
Probabilities (X = 1) (cont’d)
p̂(Dissatisfied vs. Neutral/Satisfied)
e .

 .
 .
 e
p̂(Dissatisfied/Neutral vs.Satisfied)
e .

 .
 .
 e
Cumulative Probabilities
Effect of Information on
Satisfaction (Cumulative
Probabilities)
.60
.50
.40
.30
.20
.10
.00
(0) Insufficient
(1) Sufficient
Dissatisfied
0.29
0.11
Dissatisfied/
Neutral
0.56
0.29
Information
Odds Ratio
Reverse the sign of the slope and
exponentiate it.
 e.g., OR equals .31, calculated as e-1.139
 In contrast to binary logistic regression, in
which odds are calculated as a ratio of
probabilities for higher to lower values of
the DV (odds of 1 vs. 0), in ordinal logistic
regression it is the reverse

Odds Ratio (cont’d)

SPSS reports the exponentiated slope
(e1.139= 3.123)--the sign of the slope is not
reversed before it is exponentiated (e-1.139
= .320)
Question & Answer

Are foster mothers who report that they were
provided sufficient information about the role of
foster care workers more satisfied with their
foster care agencies?

The odds of being dissatisfied (vs. neutral or
satisfied) are .32 times smaller (68%) for mothers
who received sufficient information. Similarly, the
odds of dissatisfied or neutral (vs. satisfied) are
.32 times smaller (68%) for mothers who received
sufficient information.
Single (Quantitative) IV Example

DV = satisfaction with foster care agencies
 (1) dissatisfied; (2) neither satisfied nor
dissatisfied; (3) satisfied

IV = available time to foster (Available
Time Scale); higher scores indicate more
time to foster
 Converted to z-scores

N = 300 foster mothers
Single (Quantitative) IV Example
(cont’d)

Are foster mothers with more time to
foster more satisfied with their foster
care agencies?
Statistical Significance

Table 4.3
(zTime) = 0
• Reject
Odds Ratio

OR equals .76 (e-.281)
 For a one standard-deviation increase in
available time, the odds of being dissatisfied
(vs. neutral or satisfied) decrease by a factor
of .76 (24%). Similarly, for one standarddeviation increase in available time the odds of
being dissatisfied or neutral (vs. satisfied)
decrease by a factor of .76 (24%).
Figures

zATS.xls
Estimated Cumulative Logits
L (Dissatisfied vs. Neutral/Satisfied) = t1 - BX
L (Dissatisfied/Neutral vs. Satisfied) = t2 – BX
Table 4.3
L (Dissatisfied vs. Neutral/Satisfied) = -1.365 – .281X
L (Dissatisfied/Neutral vs. Satisfied) = -.269 – .281X
Effect of Time on Satisfaction
(Cumulative Logits)
1.00
Logits
0.00
-1.00
-2.00
-3.00
-3
-2
-1
0
1
2
3
Dissatisfied
-0.52
-0.80
-1.08
-1.36
-1.65
-1.93
-2.21
Dissatisfied/Neutral
0.57
0.29
0.01
-0.27
-0.55
-0.83
-1.11
Available Time to Foster
Effect of Time on Satisfaction
(Cumulative Odds)
2.00
Odds
1.50
1.00
0.50
0.00
-3
-2
-1
0
1
2
3
Dissatisfied
0.59
0.45
0.34
0.26
0.19
0.15
0.11
Dissatisfied/Neutral
1.77
1.34
1.01
0.76
0.58
0.44
0.33
Available Time to Foster
Effect of Time on Satisfaction
(Cumulative Probabilities)
Cumulative Probabilities
.70
.60
.50
.40
.30
.20
.10
.00
-3
-2
-1
0
1
2
3
Dissatisfied
0.37
0.31
0.25
0.20
0.16
0.13
0.10
Dissatisfied/Neutral
0.64
0.57
0.50
0.43
0.37
0.30
0.25
Available Time to Foster
Question & Answer

Are foster mothers with more time to foster more
satisfied with their foster care agencies?

For a one standard-deviation increase in available
time, the odds of being dissatisfied (vs. neutral or
satisfied) decrease by a factor of .76 (24%).
Similarly, for one standard-deviation increase in
available time the odds of being dissatisfied or
neutral (vs. satisfied) decrease by a factor of .76
(24%).
Multiple IV Example


DV = satisfaction with foster care agencies
 (1) dissatisfied; (2) neither satisfied nor
dissatisfied; (3) satisfied
IV = available time to foster (Available Time
Scale); higher scores indicate more time to foster
 Converted to z-scores
IV = agencies provided sufficient information
about the role of foster care workers
 0 (no) or 1 (yes)
 N = 300 foster mothers

Multiple IV Example (cont’d)

Are foster mothers who receive sufficient
information about the role of foster care
workers more satisfied with their foster
care agencies, controlling for available time
to foster?
Statistical Significance

Table 4.4
 (Info) = (zTime) = 0
• Reject

Table 4.5
 (Info) = 0
• Reject
 (zTime) = 0
• Reject

Table 4.6
 (Info) = 0
• Reject
 (zTime) = 0
• Reject
Odds Ratio: Information

OR equals .33 (e-1.116)
 The odds of being dissatisfied (vs. neutral or
satisfied) are .33 times (67%) smaller for
mothers who received sufficient information,
when controlling for available time to foster.
Similarly, the odds of being dissatisfied or
neutral (vs. satisfied) are .33 times (67%)
smaller for mothers who received sufficient
information, when controlling for time.
Odds Ratio: Time

OR equals .77 (e-.260)
 For a one standard-deviation increase in
available time, the odds of being dissatisfied
(vs. neutral or satisfied) decrease by a factor
of .76 (24%), when controlling for information.
Similarly, for one standard-deviation increase in
available time the odds of being dissatisfied or
neutral (vs. satisfied) decrease by a factor of
.76 (24%), when controlling for information.
Estimated Cumulative Logits
Table 4.6
 L(Dissatisfied vs. Neutral/Satisfied) =
-.941 – [(1.116)(XInfo) + (.260)(XzTime)]
L(Dissatisfied/Neutral vs. Satisfied) =
.222 – [(1.116)(XInfo) + (.260)(XzTime)]

Estimated Odds as a Function of
Available Time and Information

See Table 4.7
Estimated Probabilities as a
Function of Available Time and
Information

See Table 4.9
Question & Answer

Are foster mothers who receive sufficient
information about the role of foster care workers
more satisfied with their foster care agencies,
controlling for available time to foster?

The odds of being dissatisfied (vs. neutral or
satisfied) are .33 times (67%) smaller for mothers
who received sufficient information, when
controlling for available time to foster. Similarly,
the odds of being dissatisfied or neutral (vs.
satisfied) are .33 times (67%) smaller for mothers
who received sufficient information, when
controlling for time.
Assumptions Necessary for
Testing Hypotheses


Assumptions discussed in GZLM lecture
Effect of the IVs is the same for all values of the
DV (“parallel lines assumption”)
L(Dissatisfied vs. Neutral/Satisfied) = t1 – (BInfoXInfo + BzTimeXzTime)
L(Dissatisfied/Neutral vs. Satisfied) = t2 - (BInfoXInfo + BzTimeXzTime)


Ordinal logistic regression assumes that BInfo is
the same for both equations, and BzTime is the same
for both equations
See Table 4.10
Model Evaluation

Create a set of binary DVs from the
polytomous DV
compute Satisfaction (1=1) (2=0) (3=0) into SatisfactionLessThan2.
compute Satisfaction (1=1) (2=1) (3=0) into SatisfactionLessThan3.
Run separate binary logistic regressions
 Use binary logistic regression methods to
detect outliers and influential observations

Model Evaluation (cont’d)

Index plots
 Leverage values
 Standardized or unstandardized deviance
residuals
 Cook’s D

Graph and compare observed and
estimated counts
Analogs of R2
None in standard use and each may give
different results
 Typically much smaller than R2 values in
linear regression
 Difficult to interpret

Multicollinearity
SPSS GZLM doesn’t compute
multicollinearity statistics
 Use SPSS linear regression
 Problematic levels

 Tolerance < .10 or
 VIF > 10
Additional Topics
Polytomous IVs
 Curvilinear relationships
 Interactions

Additional Regression Models for
Polytomous DVs

Ordinal probit regression
 Substantive results essentially indistinguishable
from ordinal logistic regression
 Choice between this and ordinal logistic
regression largely one of convenience and
discipline-specific convention
 Many researchers prefer ordinal logistic
regression because it provides odds ratios
whereas ordinal probit regression does not, and
ordinal logistic regression comes with a wider
variety of fit statistics
Additional Regression Models for
Polytomous DVs (cont’d)

Adjacent-category logistic model
 Compares each value of the DV to the next
higher value

Continuation-ratio logistic model
 Compares each value of the DV to all lower
values

Generalized ordered logit model
 Relaxes the parallel lines assumption
Additional Regression Models for
Polytomous DVs (cont’d)

Complementary log-log link (also known as
clog-log)
 Useful when higher categories more probable

Negative log-log link
 Useful when lower categories more probable

Cauchit link
 Useful when DV has a number of extreme
values