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Probit and Logit


Linear Probability Response Models

Probit and Logit

Techniques for analyzing the relationship between fixed-level independent variables and a dependent variable constrained to vary between 0 and 1

Key Concepts

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Probit & Logit

The analytic differences between linear regression and the linear probability

models Probit and Logit

Fixed v. random variable

Constrained v. unconstrained variables

The analytic differences between logistical regression and Probit & Logit

The definition of a Probit transformation

The definition of a Logit transformation

Assumptions of Probit and Logit

Maximum likelihood estimation and Probit/Logit parameters

The meaning of linearity in Probit and Logit analysis

Interpretation of the constant (a) and regression coefficient (b) in Probit analysis

Interpretation of the constant (a) and regression coefficient (b) in Logit analysis

Chi square goodness-of-fit test in Probit and Logit and its null hypothesis

Expected residual scatterplot if Probit/Logit a and b are not significant

The concept of a residual in Probit and Logit

The concept of median potency

Back-solving for the median potency from a Probit or Logit equation

95% confidence interval of the median potency

The purpose and method of logarithmic transformation in Probit and Logit

analysis

Common v. natural logarithms

Logarithms and antilogarithms

Laws of logarithms

Bivariate Probit and Logit models

Multivariate Probit and Logit models

Probit and Logit models with multiple groups and/or multiple IV(s)

The concept of a common slope in Probit/Logit multiple group analysis

Parallelism test for the homogeneity of regression slopes for factor levels

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Lecture Outline

ü  An example: predicting success of juvenile probationers

ü  The theory of the Probit Response Model

ü  Improving the goodness of fit of a Probit model with logarithmic transformation

ü  Interpreting a log transformed Probit model

ü  The theory of the Logit Response Model

ü  Comparison of Probit & Logit Response Models

ü  Probit with two independent variables

ü  Probit with multiple groups

ü  Probit with multiple independent variables & multiple groups


Analytic Problem

Linear regression assumes

That there are no restrictions on the values

of the IV(s), that the values are random

That the DV is continuous

And that since the IV(s) can take on any

value, the DV can range from ± ¥

The problem

What if the IV(s) has fixed levels, vis-à-vis

being random?

What if the DV is constrained to be between

0.0 & 1.0 (as in a proportion)?

Use of linear regression would lead to an

egregious violation of assumptions

The solution

Use  a linear probability model such as

Probit or Logit


Database: Juvenile Probation 1

Probation Outcome Among Juvenile Shoplifters as a Function of Months of Supervision

Succeeded

Failed
Months N p N p Total N
6 18 .47 21 .53 38
7 23 .55 19 .45 42
8 22 .61 14 .39 36
9 19 .63 11 .37 30
10 32 .66 16 .34 48
11 32 .71 13 .29 45
12 30 .75 10 .25 40
Totals 176 103 279

N = number of juveniles, p = proportion of successful juveniles

Independent variable (metric & fixed): months of supervision

Dependent variable: the proportion that succeeded

Overall success = (176) / (279) = 0.63, or 63%


Research Questions

  1. Is there a significant relationship between success on probation and the number of months of supervision?
  1. Analytic approaches:

Series of t-tests comparing differences in proportions of successes for each of seven supervision groups

Chi square analysis of months by success/failure

Logistical regression: success/failure (0/1) as a function of months

  1. By how much does the proportion of successes increase for each one month of supervision?
  1. How much supervision was required to achieve a 50% success rate, 75%, 90%, etc.?


Research questions ( con’d )

  1. Multivariate questions:

Does the effect of the number of months of supervision on success vary with the age of the juvenile?

Does the effect of the number of months supervision on success vary with the age and gender of the juvenile?


Probit: Linear Probability Model

Transform (Pi) = a + bXI

Dependency Technique

One or more metric IV(s) reduced to fixed

levels or “doses”

One binary DV, measured as the proportion

of the subjects responding to a fixed level of

the IV. DV is transformed to Z score values.

Parameters are estimated by maximum likelihood estimation, not OLS.

Assumptions

The relationship between the levels of the

IV(s) and the DV is linear

Homogeneity of variance of the DV over

levels of the IV

Homoskedastic residuals

Absence of outliers


The Probit Model

Transform (Pi) = a + bXI

PI = Probit: the Z score associated with the proportion of cases responding to the stimulus (e.g. proportion successful on probation)

a = constant

b = Probit regression coefficient

Xi = Level of the independent variable (e.g. months of supervision)

Probit transformation: Convert the proportion responding to a level of the IV to the corresponding Z score on a standard normal curve

Example: 22 out of 36 juveniles succeeded with 8 months supervision. (22) / (36) = 0.611

Z = + 0.28 (0.611 is approximately the proportion in the larger area of the curve associated with a Z score of +0.28)

Insert a standard normal probability table with values of Z from 0.0 to 0.40

Probit Transformation

22 out of 36 juveniles succeeded

with 8 months of supervision (61.1%)

Standard Normal Curve

Area = 0.611

Z = 0.00

Z = + 0.28


Probit & Logit Case Studies of Probation Outcome Study

Model 1            Probit response model

Success = f (months of supervision)

Model 2            Log transform Probit model

Success = f (log months)

Model 3            Logit response model

Success = f (months of supervision)

Model 4            Log transform Logit model

Success = f (log months)

Model 5            Probit response model

Success = f (months & age)

Model 6            2 Probit response models

Male Success = f (months)

Female Success = f (months)

Model 7            2 Probit multivariate models

Male Success = f (months & age)

Female Success = f (months & age)


Model 1

The Effect of the Number of Months of Supervision on Probation Success

Probit = a + b (months)

Probit = -0.70946 + 0.11582 (months)


Probit Model of the Effect of Months of Supervision on Probation Success

Months of Supervision Total Number of Juveniles Number Successful
6 38 18 (0.47)
7 42 23 (0.55)
8 36 22 (0.61)
9 30 19 (0.63)
10 48 32 (0.67)
11 45 32 (0.71)
12 40 30 (0.75)

Values in parentheses are the proportions of juveniles who were successful.


Conversion of the Proportion Successful to Associated Z Scores

Months of Supervision Total Number of Juveniles Number Successful (p) Probit Z Score
6 38 18 (0.47) -0.08
7 42 23 (0.55) +0.13
8 36 22 (0.61) +0.28
9 30 19 (0.63) +0.33
10 48 32 (0.67) +0.44
11 45 32 (0.71) +0.55
12 40 30 (0.75) +0.67

Plot of the Number of Successful Probationers as a Function of

Months of Supervision

Compare this scatterplot with the Probit transform plot on the following exhibit.

Plot of Probit Z Scores as a Function of Months of Supervision

Notice that the conversion of the proportion of successful juvenile
s to Probit Z scores results in a linear relationship over time.

As months of supervision increases, the value of the Probit increases, i.e. the greater the likelihood of being successful on probation.

In Probit analysis, the equation of the best fit line is achieved by maximum likelihood estimation.


What is the Linear Relationship Between Months of Supervision and  Probation Success?

The Probit model

Probit = a + b (months)

Estimated best fit linear equation

Probit = -0.70946 + 0.11582 (months)

Interpretation

For every one-month increase in the amount of supervision, the Probit of success (Z score) increases by 0.11582.


Results of the Probit Analysis

Parameter estimates converged after 9 iterations.

Optimal solution found.

Parameter Estimates (PROBIT model:  (PROBIT(p)) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTHS             .11582          .03862         2.99914

Intercept  Standard Error  Intercept/S.E.

-.70946          .35511        -1.99788

Pearson  Goodness-of-Fit  Chi Square =       .180    DF = 5   P =  .999

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Probit Model

Probit = a + b (month)

a = intercept = -0.70946 (t = -1.99788, df = N – k = 5, p< 0.05)

b = Probit regression coefficient = 0.11582

(t = 2.99914, df = N – k = 5, p< 0.05)

N = 7 months, k = 2 = number of parameters estimated (a & b)

Probit Model Predictions of the Effect of Months of Supervision on Success

Probit = -0.70946 + 0.11582 (months)

Example: Predicted success of juveniles with 8 months supervision

Total = 36, 22 succeeded or 61.1%

Probit = -0.70946 + 0.11582 ( 8 months)

Probit = +0.2171

Z of +0.2171 has an associated area under

the larger portion of the curve of 0.5859

(36 juveniles) (0.5859) = 21.09 successes

Residual = (22 – 21.09) = 0.91


Probit Transformation

22 out of 36 juveniles succeeded with 8 months of supervision (61.1%). The Probit equation yields a Z = +0.2171. Since Z is positive, the area in the larger portion of the curve is 0.5859, or a prediction of a 58.59 % success rate.

Standard Normal Curve

Area = 0.5859

Z = 0.00

Z = + 0.2171


Scatterplot of the Residuals Produced By the Probit Model

Probit Residuals

0.00

1          2          3          4          5          6          7          8          9          10

Months of Supervision


Chi Square Goodness of Fit Test

l2 = S { (residual)2 / [ ni Pi (1-Pi) ] }

Residual = (observed – expected successes)

nI = Number of cases in the group, i.e. level of IV

PI = Predicted proportion of cases (cf. the next exhibit for a summary of the predictions and the residuals)

l2 = (-0.78)2 / [(38)(0.4942)(1-0.4942)] +

(0.36)2 / [(42)(0.54034)(1-0.54034)] + … +

(-0.075)2 / [(40)(0.75187)(1-0.75187)] = 0.18

l2 = 0.18, df = (g-k) = (7-2) = 5, p = 0.999

g = number of supervision groups, k = number of parameters estimated (i.e. 2, a & b)

Decision: Accept H0, no difference between the observed and predicted proportion of successes. The Probit model fits the data.

Results of the Chi-Square Analysis of the Residuals

Observed and Expected Frequencies

Number of    Observed    Expected

MONTHS   Subjects   Responses   Responses    Residual     Prob

6.00       38.0        18.0      18.780       -.780   .49420

7.00       42.0        23.0      22.694        .306   .54034

8.00       36.0        22.0      21.094        .906   .58593

9.00       30.0        19.0      18.912        .088   .63040

10.00       48.0        32.0      32.313       -.313   .67319

11.00       45.0        32.0      32.122       -.122   .71381

12.00       40.0        30.0      30.075       -.075   .75187

Pearson  Goodness-of-Fit  Chi Square =       .180    DF = 5   P =  .999

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Prob = expected probability of success

Prob = (expected responses / number of subjects)

The null hypothesis that there is no significant difference between the observed and expected number of successful probationers is accepted.

Decision The Probit model fits the data.

Table of Probit Model Predictions

Confidence Limits for Effective MONTHS

95% Confidence Limits

Prob        MONTHS         Lower         Upper

.01     -13.96029     -57.27643      -4.83864

.02     -11.60665     -50.48995      -3.40871

.03     -10.11335     -46.18498      -2.50064

.04      -8.98999     -42.94706      -1.81699

.05      -8.07623     -40.31367      -1.26048

.06      -7.29847     -38.07258       -.78647

.07      -6.61653     -36.10789       -.37055

.08      -6.00593     -34.34901        .00213

.09      -5.45062     -32.74962        .34131

.10      -4.93945     -31.27762        .65376

.15      -2.82309     -25.18616       1.95040

.20      -1.14107     -20.34978       2.98586

.25        .30196     -16.20613       3.87972

.30       1.59784     -12.49186       4.68929

.35       2.79866      -9.05938       5.44882

.40       3.93813      -5.81619       6.18344

.45       5.04057      -2.70143       6.91725

.50       6.12554        .32014       7.68324

.55       7.21051       3.24192       8.54902

.60       8.31296       5.93250       9.70702

.65       9.45242       7.99500      11.62233

.70      10.65325       9.32416      14.48518

.75      11.94913      10.35800      17.97518

.80      13.39215      11.35755      22.01314

.85      15.07417      12.45162      26.79091

.90      17.19054      13.78504      32.84560

.91      17.70171      14.10331      34.31178

.92      18.25702      14.44795      35.90570

.93      18.86762      14.82578      37.65943

.94      19.54956      15.24661      39.61921

.95      20.32731      15.72535      41.85558

.96      21.24108      16.28646      44.48435

.97      22.36443      16.97471      47.71768

.98      23.85774      17.88757      52.01787

.99      26.21138      19.32293      58.79891

Median potency

Number of months of supervision which yeilds a Probit Z score = 0.0, 50% success rate, 6.12554 months

Probit = -0.70946 + 0.11582 (month)

0.0 = -0.70946 + 0.11582 (month)

month = (0.70946 / 0.11582) = 6.12554

Model 2

Logarithmic Transformed Probit Model of the Effect of Months of Supervision on Probation Success

Probit = a + b (log months)

Probit = -1.85282 + 2.31804 (log months)

Logarithmic Transformation of Probit

Probit assumes a linear relationship between the IV and the Z score transformation of the DV.

Sometimes the goodness of fit of a Probit model can be improved by a logarithmic transformation of the independent variable (s), e.g. months.

Model 2: Log transformed Probit model

A common logarithmic transformation is used, i.e. base 10.

Probit = a + b ( log months )

Probit = -1.85282 + 2.31804 (  log months )

What is a Logarithm?

The logarithm of a number (N) is the exponent (x) to which a base number (b) must be raised to yield that number (N).

Log b N = x             (b) X = N

Common logarithms

Base 10 (symbolized log)

Log 6 = 0.77815

(10)0.77815 = 5.9999 @ 6.0

Natural logarithms

Base  e, e = 2.71828 (symbolized ln)

ln 6 = 1.791759

(e)1.791759 = 5.9999 @ 6.0

The Base of the Natural Logarithms (e)

The base of the natural logarithms is defined as the limit of the following function (e = 2.71828):

e = lim  (1 + 1/u)u

uÙ¥

U

(1 + 1/u)u

1 2.00000
10 2.59374
100 2.70481
1000 2.71692
10,000 2.71815
100,000 2.71827
1,000,000 2.71828 = e

Algebraic Laws of Logarithms

The example uses common logarithms, but the same laws apply to natural logarithms.

1st Law: law of multiplication

log (3 x 2) = log 3 + log 2

log (6) = (90.47712 + 0.30103)

0.77815 = 0.77815

2nd Law: law of division

log (3 / 2) = log 3 – log 2

log 1.5 = (0.47712 – 0.30103)

0.17609 = 0.17609

3rd Law: law of exponents

log (3)2 = 2 (log 3)

log (9) = 2 (0.47712) = 0.95424

Logarithmic Transformation of

Months of Supervision

The objective of the transformation is to linearize the relationship between months of supervision and the proportion of successes.

Two options

Common logarithmic transformation (log)

Natural logarithmic transformation (ln)

Months Log

Base 10

ln

Base e

6 0.778 1.792
7 0.845 1.946
8 0.903 2.079
9 0.954 2.197
10 1.000 2.303
11 1.041 2.398
12 1.079 2.485

Plot of the Z Transformed Proportion of Successful Juvenile Probationers as a Function of the Log 10 of Months of Supervision

Compare this with the plot of the Probit as a function of months of supervision in Model 1. (cf. p. 17)

Is the relationship more graphically linear as a result of the transformation of the number of months of supervision to the log of months?

Results of the Log 10 Transfomed

Probit Analysis

Parameter estimates converged after 10 iterations.

Optimal solution found.

Parameter Estimates (PROBIT model:  (PROBIT(p)) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTHS            2.31804          .76881         3.01510

Intercept  Standard Error  Intercept/S.E.

-1.85282          .72824        -2.54425

Pearson  Goodness-of-Fit  Chi Square =       .105    DF = 5   P = 1.000

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Probit = -1.85282 + 2.31804 (log month)

Prediction of success after 8 months

Probit = -1.85282 + 2.31804 (log 8)

Probit = -1.85282 + 2.31804 (0.903089)

= +0.240579

A Z of +0.240579 has a larger area of 0.59506

Therefore the predicted number of successes equals (36 total cases) (0.59506 successes) = 21.402 successful cases

Analysis of the Residuals

Observed and Expected Frequencies

Number of    Observed    Expected

MONTHS   Subjects   Responses   Responses    Residual     Prob

.78       38.0        18.0      18.257       -.257   .48045

.85       42.0        23.0      22.775        .225   .54227

.90       36.0        22.0      21.422        .578   .59506

.95       30.0        19.0      19.208       -.208   .64026

1.00       48.0        32.0      32.597       -.597   .67911

1.04       45.0        32.0      32.070       -.070   .71266

1.08       40.0        30.0      29.670        .330   .74175

Pearson  Goodness-of-Fit  Chi Square =       .105    DF = 5   P = 1.000

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Test of goodness of fit

Chi-square = 0.105, p = 1.00

There is no significant difference between the actual and predicted number of successes.

Decision The model fits the data.

Table of Probit Model Predictions

Confidence Limits for Effective MONTHS

95% Confidence Limits

Prob        MONTHS         Lower         Upper

.01        .62477        .00461       1.77880

.02        .81907        .00999       2.09760

.03        .97259        .01631       2.32911

.04       1.10678        .02358       2.52012

.05       1.22946        .03183       2.68711

.06       1.34455        .04109       2.83805

.07       1.45429        .05140       2.97746

.08       1.56012        .06280       3.10818

.09       1.66304        .07535       3.23213

.10       1.76378        .08910       3.35068

.15       2.25003        .17833       3.89097

.20       2.73042        .30936       4.38435

.25       3.22352        .49595       4.86029

.30       3.74178        .75713       5.33580

.35       4.29613       1.11933       5.82412

.40       4.89791       1.61946       6.33891

.45       5.56025       2.30899       6.89853

.50       6.29948       3.25719       7.53533

.55       7.13698       4.54260       8.32544

.60       8.10211       6.17111       9.50913

.65       9.23701       7.81138      11.82974

.70      10.60548       9.10228      16.38213

.75      12.31059      10.25491      24.37005

.80      14.53380      11.50799      38.59418

.85      17.63684      13.05549      66.50014

.90      22.49907      15.22534     132.52929

.91      23.86189      15.79445     156.61674

.92      25.43613      16.43471     187.79510

.93      27.28722      17.16649     229.31689

.94      29.51429      18.01996     286.66982

.95      32.27697      19.04257     369.83707

.96      35.85492      20.31525     498.94425

.97      40.80159      21.99299     721.10866

.98      48.44956      24.43386    1176.86871

.99      63.51682      28.83116    2547.90778

Median potency

50% success rate is predicted at 6.29948 months of supervision

Model 1 predicted a median potency at 6.29948 months

Comparison of Models 1 & 2

The Effect of Log10 Transformation of Months of Supervision

Transformation
Statistic Model 1

None

Model 2

Log10

a (the constant) -0.70946 -1.85282
ta -1.99788 -2.54425
b (Probit coefficient) +0.11582 +2.31804
tb +2.99914 +3.01510
l2 (goodness of fit) 0.180 0.105
p of l2 (significance) 0.999 1.000
S (residuals)2 1.649 0.964
Median Potency (MP)

(in months)

6.126 6.299

The log transform model produced a slightly more significant fit and slightly lower residuals. However the operational difference between the two models is negligible.

Prediction With the Log Transformed

Probit Model

Probit = -1.85282 + 2.31804 (log months)

Example:: Expected success after 8 months of supervision.

Probit = -1.85282 + 2.31804 (log 8)

Probit = -1.85282 + 2.31804 (0.903)

Probit = +0.2406

Z score of +0.2406 divides a standard normal

curve into two parts corresponding to 0.5951 &

0.4049

area = 0.5951

area = 0.4049

Given 36 juveniles

(36) (0.5951) = 21.42

21 juveniles expected to

be successful

Z = +0.2406

Interpretation of the Regression Coefficient (b) in the Probit Log Transformed Model

Probit = a + b (log months)

Probit = -1.85282 + 2.31804 (log months)

Interpretation: For every unit increase in the

log month

The Probit (Z score) increases

by 2.31804

Month Log Month Probit Expected Successful
10 1.0 0.46522 0.681
100 2.0 2.78326 0.997
Difference = 2.31804

Model 3

Logit Model of the Effect of Months of Supervision on Probation Success

Logit = a + b (months)

Logit = -1.15774 + 0.18841 (month)

Logit: An Alternative Linear

Probability Model

Transform (Li) = a + bXI

Dependency Technique

One or more metric IV(s) reduced to fixed

levels or “doses”

One binary DV, measured as the proportion

of the subjects responding to a fixed level of

the IV. DV is transformed to ln [Pi / (1-Pi)].

Parameters are estimated by maximum likelihood estimation.

Assumptions

That there is a linear relationship between

levels of the IV(s) and the DV

Homogeneity of variance of the DV over

levels of the IV

Homoskedastic residuals

Absence of outliers

The Logit Model

Transform (Li) = a + bXI

LI = Logit = ln [Pi / (1-Pi)],

ln = natural logarithm

Pi = the proportion of subjects in a group effected by a level of the IV (e.g. proportion successful)

a = constant

b = Logit regression coefficient

Xi = Level of the independent variable (e.g. months of supervision)

Example: 22 out of 36 succeeded with 8 months supervision

(22) / (36) = 0.611

Logit = ln [0.6111 / (1 – 0.6111)]  = 0.447

Comparison of Probit and Logit Transformations

Month Total

Juveniles

Number Success P

Success

Probit

(Z score)

Logit

ln [Pi / (1-Pi)]

6 38 18 0.47 -0.07 -0.120
7 42 23 0.55 +0.13 +0.201
8 36 22 0.61 +0.28 +0.447
9 30 19 0.63 +0.33 +0.532
10 48 32 0.66 +0.44 +0.663
11 45 32 0.71 +0.55 +0.895
12 40 30 0.75 +0.68 +1.099

Plot of the Logit as a Function of

Months of Supervision

Notice how the logit transformation of the proportion of successes has linearized the relationship between success and months of supervision.

Results of the Logit Analysis of Probation Success as a Function of Months of Supervision

Parameter estimates converged after 8 iterations.

Optimal solution found.

Parameter Estimates (LOGIT model:  (LOG(p/(1-p))) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTHS             .18841          .06326         2.97832

Intercept  Standard Error  Intercept/S.E.

-1.15774          .57656        -2.00802

Pearson  Goodness-of-Fit  Chi Square =       .171    DF = 5   P =  .999

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Logit = -1.15774 + 0.18841 (month)

Predicted success with 8 months supervision

Logit = ln (P / 1-P) = -1.15774 + 0.18841 (8)

Results of the Logit Analysis of Probation Success as a Function of Months of Supervision (cont.)

ln (P / 1-P) = 0.34954

Taking the antilog of both sides of the equation

(P / 1 – P) = e 0.34954 = 2.71828 0.34954 = 1.4184

(P / 1 – P) = 1.4184

P = 1.4184 (1 – P) = 1.4184 – 1.4184 P

0 = 1.4184 – 1.4184 P – P = 1.4184 – 2.4184 P

P = 1.4184 / 2.4184 = 0.5865

With 8 months of supervision, it is predicted that 0.5865 proportion, or 58.65%, of the juveniles will be successful.

Given 36 juveniles supervised for 8 months, the model predicts that 21.11 will be successful

(36 juveniles) (0.5865 successful) = 21.11

Residual Analysis

Observed and Expected Frequencies

Number of    Observed    Expected

MONTHS   Subjects   Responses   Responses    Residual     Prob

6.00       38.0        18.0      18.741       -.741   .49318

7.00       42.0        23.0      22.688        .312   .54020

8.00       36.0        22.0      21.114        .886   .58651

9.00       30.0        19.0      18.940        .060   .63134

10.00       48.0        32.0      32.353       -.353   .67401

11.00       45.0        32.0      32.129       -.129   .71398

12.00       40.0        30.0      30.034       -.034   .75086

Pearson  Goodness-of-Fit  Chi Square =       .171    DF = 5   P =  .999

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

The chi-square analysis indicates that there is not a significant difference between the actual and predicted number of successes.

Decision The Logit model fits the data.

Table of Logit Model Predictions

Confidence Limits for Effective MONTHS

95% Confidence Limits

Prob        MONTHS         Lower         Upper

.01     -18.24389     -70.52373      -7.42514

.02     -14.51112     -59.61405      -5.16623

.03     -12.30468     -53.16658      -3.82971

.04     -10.72281     -48.54497      -2.87072

.05      -9.48290     -44.92304      -2.11845

.06      -8.45906     -41.93276      -1.49679

.07      -7.58414     -39.37783       -.96515

.08      -6.81804     -37.14104       -.49926

.09      -6.13490     -35.14680       -.08350

.10      -5.51705     -33.34345        .29283

.15      -3.06168     -26.18059       1.79213

.20      -1.21304     -20.79354       2.92679

.25        .31383     -16.35043       3.87024

.30       1.64769     -12.47658       4.70201

.35       2.85917      -8.96820       5.46755

.40       3.99271      -5.70029       6.19860

.45       5.07966      -2.59097       6.92386

.50       6.14472        .40982       7.68044

.55       7.20978       3.30559       8.54203

.60       8.29673       5.96602       9.71617

.65       9.43027       7.98983      11.69133

.70      10.64176       9.30236      14.65272

.75      11.97561      10.35044      18.31026

.80      13.50248      11.39645      22.65080

.85      15.35112      12.58812      27.98084

.90      17.80650      14.12312      35.10802

.91      18.42434      14.50507      36.90573

.92      19.10748      14.92610      38.89471

.93      19.87358      15.39694      41.12655

.94      20.74850      15.93328      43.67678

.95      21.77234      16.55944      46.66256

.96      23.01225      17.31607      50.28014

.97      24.59412      18.27936      54.89744

.98      26.80057      19.62028      61.34050

.99      30.53333      21.88407      72.24531
Model 4

Logarithic Transformed Logit Model of the Effect of Months of Supervision on Probation Success

Logit = a + b (log month)

Logit = -3.00611 + 3.75813 (log month)

Plot of the Logits as a Function of the Log Transformed Months of Supervistion

Compare this scatterplot with the plot associated with Model 3. (cf. p. 43)

Did the log transformation of months improve the linearity of the relationship between the logit and months of supervision?

Results of the Logit Log Transformed Analysis

Parameter estimates converged after 13 iterations.

Optimal solution found.

Parameter Estimates (LOGIT model:  (LOG(p/(1-p))) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTHS            3.75813         1.25339         2.99836

Intercept  Standard Error  Intercept/S.E.

-3.00611         1.18264        -2.54186

Pearson  Goodness-of-Fit  Chi Square =       .108    DF = 5   P = 1.000

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Logit = -3.00611 + 3.75813 (log month)

Prediction with Logarithic Transformed Logit Model

Logit = -3.00611 + 3.75813 (log month)

Example: Predicted success of juveniles with 8 months supervision. Total = 36, of which 22 succeeded or 61.1%

Logit = ln [Pi / (1-Pi)]

ln [Pi / (1-Pi)] = -3.00611+ 3.75813 ( log 8 mos)

ln [Pi / (1-Pi)] = +0.3878

Take the antilog of both sides of the equation

Pi / (1-Pi) = e 0.3878 = (2.71828)0.3878 = 1.4737

P = 0.5957 proportion successful

(36 juveniles) (0.5957) = 21.45 successes

Residual Analysis

Observed and Expected Frequencies

Number of    Observed    Expected

MONTHS   Subjects   Responses   Responses    Residual     Prob

.78       38.0        18.0      18.224       -.224   .47958

.85       42.0        23.0      22.779        .221   .54237

.90       36.0        22.0      21.447        .553   .59576

.95       30.0        19.0      19.232       -.232   .64108

1.00       48.0        32.0      32.622       -.622   .67962

1.04       45.0        32.0      32.063       -.063   .71250

1.08       40.0        30.0      29.628        .372   .74070

Pearson  Goodness-of-Fit  Chi Square =       .108    DF = 5   P = 1.000

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

The chi-square analysis indicates that there is not a significant difference between the actual and predicted number of successes.

Decision The Logit model fits the data.

Table of Logit Model Predictions

Confidence Limits for Effective MONTHS

95% Confidence Limits

Prob        MONTHS         Lower         Upper

.01        .37773        .00101       1.31099

.02        .58119        .00351       1.70275

.03        .74979        .00733       1.98764

.04        .90001        .01241       2.22098

.05       1.03851        .01875       2.42304

.06       1.16880        .02636       2.60382

.07       1.29302        .03527       2.76908

.08       1.41258        .04551       2.92251

.09       1.52848        .05712       3.06659

.10       1.64148        .07016       3.20312

.15       2.17940        .15871       3.81016

.20       2.69785        .29324       4.34493

.25       3.21786        .48656       4.84630

.30       3.75352        .75659       5.33610

.35       4.31694       1.12847       5.83053

.40       4.92047       1.63760       6.34536

.45       5.57828       2.33380       6.90093

.50       6.30807       3.28497       7.53211

.55       7.13334       4.56867       8.32023

.60       8.08699       6.18747       9.52183

.65       9.21758       7.80398      11.92296

.70      10.60118       9.08367      16.69367

.75      12.36592      10.25675      25.31287

.80      14.74945      11.57796      41.50041

.85      18.25815      13.29108      76.17175

.90      24.24132      15.87592     171.59932

.91      26.03346      16.59364     210.61476

.92      28.16962      17.42242     264.19811

.93      30.77436      18.39837     340.71509

.94      34.04496      19.57686     455.63122

.95      38.31622      21.04849     640.31575

.96      44.21252      22.97513     967.04695

.97      53.07036      25.68524    1636.75927

.98      68.46545      29.99797    3411.08530

.99     105.34466      38.98445   11820.75520

Comparison of Models 1, 2, 3, & 4

Type of

Model

Probit Logit
Statistic Model 1 Model2

(log)

Model 3 Model 4

(log)

b 0.11582 2.31804 0.18841 3.75813
t 2.99914 3.0151 2.97832 2.998
l2 0.180 0.105 0.171 0.108
(p) 0.999 1.000 0.999 1.000
S (residuals)2 1.649 0.964 1.577 0.988
Median Potency

(in months)

6.1255 6.2995 6.1447 6.3081

All four models produce simmilar results. The log transform models produce the smaller residuals.

The Probit log model produces the smallest sum of squared residuals and is the best of the four models.

Operationally, there is negligible difference among the four models

Model 5

A Multivariate Probit Model with Two

Independent Variables

*****

Success as a Function of Months of

Supervision and Age

Probit = a + b1 (month) + b2 (age)

Probit = 7.06114 + 0.13009 (month) – 0.56226 (age)

Results of the Probit Analysis of Probation Success as a Function of Months of Supervision and Age

Parameter estimates converged after 14 iterations.

Optimal solution found.

Parameter Estimates (PROBIT model:  (PROBIT(p)) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTH              .13009          .04005         3.24841

AGE               -.56226          .10113        -5.55984

Intercept  Standard Error  Intercept/S.E.

7.06114         1.43287         4.92797

Pearson  Goodness-of-Fit  Chi Square =     26.778    DF = 39   P =  .931

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

Probit = 7.06114 + 0.13009 (month) – 0.56226 (age)

Interpretation

When supervision increases by 1 month, the probit of success increases by +0.13009.

When age increases by one year, the probit of success decreases by -0.56226.

Predictions With the Probit Equation

Probit = 7.06114 + 0.13009 (month) – 0.56226 (age)

Example: Predicted success of 14 year olds with 8 months supervision

Total = 12, 8 succeeded or 66.67%

Probit = 7.06114 + 0.13009 (8 month) –

0.56226 (14 years)

Probit = +0.23022

Z of +0.23022 has an associated area under

the larger portion of the curve of 0.591

(12 juveniles) (0.591) = 7.09 successes

Covariate Analysis

To what extent are the two predictor variables (covariates) collinear?

Correlation between age and months of supervision: r = – 0.11185

Covariance between the predictor variables: cov = – 0.00045

Covariance(below) and Correlation(above) Matrices of Parameter Estimates

MONTH        AGE

MONTH          .00160    -.11185

AGE           -.00045     .01023

Table of Probit Model Predictions

Observed and Expected Frequencies

Number of    Observed    Expected

MONTH   Subjects   Responses   Responses    Residual     Prob

6.00        7.0         3.0       4.919      -1.919   .70276

6.00        7.0         5.0       4.919        .081   .70276

6.00        6.0         2.0       2.928       -.928   .48807

6.00        6.0         4.0       2.928       1.072   .48807

6.00        6.0         2.0       1.661        .339   .27687

6.00        6.0         2.0       1.661        .339   .27687

7.00        7.0         4.0       5.223      -1.223   .74615

7.00        7.0         7.0       5.223       1.777   .74615

7.00        7.0         2.0       3.779      -1.779   .53990

7.00        7.0         5.0       3.779       1.221   .53990

7.00        7.0         2.0       2.254       -.254   .32201

7.00        7.0         3.0       2.254        .746   .32201

8.00        6.0         5.0       4.716        .284   .78597

8.00        6.0         6.0       4.716       1.284   .78597

8.00        6.0         3.0       3.546       -.546   .59106

8.00        6.0         5.0       3.546       1.454   .59106

8.00        6.0         1.0       2.220      -1.220   .36995

8.00        6.0         2.0       2.220       -.220   .36995

9.00        5.0         4.0       4.109       -.109   .82190

9.00        5.0         5.0       4.109        .891   .82190

9.00        5.0         3.0       3.204       -.204   .64071

9.00        5.0         3.0       3.204       -.204   .64071

9.00        5.0         2.0       2.100       -.100   .42000

9.00        5.0         2.0       2.100       -.100   .42000

10.00        8.0         6.0       6.830       -.830   .85376

10.00        8.0         8.0       6.830       1.170   .85376

10.00        8.0         5.0       5.505       -.505   .68809

10.00        8.0         6.0       5.505        .495   .68809

10.00        8.0         2.0       3.771      -1.771   .47138

10.00        8.0         5.0       3.771       1.229   .47138

11.00        8.0         6.0       7.052      -1.052   .88156

11.00        8.0         7.0       7.052       -.052   .88156

11.00        8.0         5.0       5.860       -.860   .73255

11.00        8.0         6.0       5.860        .140   .73255

11.00        7.0         3.0       3.663       -.663   .52324

11.00        7.0         5.0       3.663       1.337   .52324

12.00        6.0         5.0       5.432       -.432   .90539

12.00        6.0         6.0       5.432        .568   .90539

12.00        6.0         4.0       4.641       -.641   .77357

12.00        6.0         5.0       4.641        .359   .77357

12.00        8.0         4.0       4.598       -.598   .57471

12.00        8.0         6.0       4.598       1.402   .57471

Model 6

Probit Analysis With Multiple Groups

*****

Independent Variable =  months of supervision

Groups = gender, males & females

Probit male = a1 + b1 (month)

Probit female = a2 + b2 (month)

Probit male = -1.001139 + 0.11726 (month)

Probit female = -0.41904 + 0.11726 (month)

Probit Analysis With Multiple Groups

Sometimes it is of interest to compare the Probit equations for multiple groups to determine if there are significant differences in the Probit coefficients (i.e. slopes) predictor variables across groups.

Examples of different groups

Racial/ethnic groups

Offense groups

Gender groups

Q   Are there significant differences between boys and girls in the effect of the number of months of supervision on probation success?

The Concept of Testing the Parallelism of the Slopes of Multiple Group Probit Equations

Step 1

Estimate separate Probit equations for boys and girls on the effect of months of supervision on probation success.

Step 2

Conduct a chi-square test to determine whether the slopes (coefficients) of the two equations differ significantly, i.e.

Whether the probit coefficients of the two equations differ significantly.

Step 3

If the slopes do not differ significantly, compute a common slope to be used for each group

Plot of the Probits of Probation Success for Boys and Girls
Q   Is the linear relationship between of months of supervision on success the same (parallel) for male and female juvenile probationers?

Results of the Probit Analysis for

Boys and Girls

Parameter estimates converged after 8 iterations.

Optimal solution found.

Parameter Estimates (PROBIT model:  (PROBIT(p)) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTH              .11726          .03922         2.98946

Intercept  Standard Error  Intercept/S.E.  GENDER

-1.01139          .37197        -2.71905         1

-.41904          .36667        -1.14281         2

Pearson  Goodness-of-Fit  Chi Square =     42.756    DF = 39   P =  .313

Parallelism Test Chi Square =       .000    DF = 1   P = 1.000

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

The parallelism test found no significant difference between the slopes of the two equations, so a common slope was computed to be used in each equation, i.e. bC = 0.11726.

Probit male = -1.001139 + 0.11726 (month)

Probit female = -0.41904 + 0.11726 (month)

The Concept of a Common Slope

Probit male = -1.001139 + 0.11726 (month)

Probit female = -0.41904 + 0.11726 (month)

Probit Z Score

F

Female Slope = 0.117

M

0.00  :

Male Slope = 0.117

-0.42  :

-1.02  :

0          1          2          3          4          5          6          7          8          9          10

Months Of Supervision

Notice that the difference between the two groups is in the intercepts: females = – 0.42, males = – 1.0. Males have a lower potential for success than females, regardless of the amount of supervision.

Predictions from Probit Equations for Male & Female Probationers with 8 Months Supervision

Male Probationers

Total = 18, 9 successful or 50%

Probit = -1.01139 + 0.11726 (8 months)

Probit = -0.07331

Z of -0.07331 has an associated area under

the smaller portion of the curve of 0.4708

(18 juveniles) (0.4708) = 8.475 successes

Female Probationers

Total = 18, 13 successful or 72.2%

Probit = -0.41904 + 0.11726 (8 months)

Probit = +0.51904

Z of +0.51904 has an associated area under

the larger portion of the curve of 0.69817

(18 juveniles) (0.69817) = 12.567 successes

Residual Analysis

Observed and Expected Frequencies

Number of    Observed    Expected

GENDER        MONTH   Subjects   Responses   Responses    Residual     Prob

1          6.00        7.0         3.0       2.654        .346   .37911

1          6.00        6.0         2.0       2.275       -.275   .37911

1          6.00        6.0         2.0       2.275       -.275   .37911

1          7.00        7.0         4.0       2.971       1.029   .42443

1          7.00        7.0         2.0       2.971       -.971   .42443

1          7.00        7.0         2.0       2.971       -.971   .42443

1          8.00        6.0         5.0       2.825       2.175   .47078

1          8.00        6.0         3.0       2.825        .175   .47078

1          8.00        6.0         1.0       2.825      -1.825   .47078

1          9.00        5.0         4.0       2.588       1.412   .51753

1          9.00        5.0         3.0       2.588        .412   .51753

1          9.00        5.0         2.0       2.588       -.588   .51753

1         10.00        8.0         6.0       4.512       1.488   .56404

1         10.00        8.0         5.0       4.512        .488   .56404

1         10.00        8.0         2.0       4.512      -2.512   .56404

1         11.00        8.0         6.0       4.877       1.123   .60968

1         11.00        8.0         5.0       4.877        .123   .60968

1         11.00        7.0         3.0       4.268      -1.268   .60968

1         12.00        6.0         5.0       3.923       1.077   .65385

1         12.00        6.0         4.0       3.923        .077   .65385

1         12.00        8.0         4.0       5.231      -1.231   .65385

2          6.00        7.0         5.0       4.284        .716   .61200

2          6.00        6.0         4.0       3.672        .328   .61200

2          6.00        6.0         2.0       3.672      -1.672   .61200

2          7.00        7.0         7.0       4.593       2.407   .65608

2          7.00        7.0         5.0       4.593        .407   .65608

2          7.00        7.0         3.0       4.593      -1.593   .65608

2          8.00        6.0         6.0       4.189       1.811   .69814

2          8.00        6.0         5.0       4.189        .811   .69814

2          8.00        6.0         2.0       4.189      -2.189   .69814

2          9.00        5.0         5.0       3.689       1.311   .73771

2          9.00        5.0         3.0       3.689       -.689   .73771

2          9.00        5.0         2.0       3.689      -1.689   .73771

2         10.00        8.0         8.0       6.196       1.804   .77445

2         10.00        8.0         6.0       6.196       -.196   .77445

2         10.00        8.0         5.0       6.196      -1.196   .77445

2         11.00        8.0         7.0       6.465        .535   .80808

2         11.00        8.0         6.0       6.465       -.465   .80808

2         11.00        7.0         5.0       5.657       -.657   .80808

2         12.00        6.0         6.0       5.031        .969   .83845

2         12.00        6.0         5.0       5.031       -.031   .83845

2         12.00        8.0         6.0       6.708       -.708   .83845

Table of Probit Model Predictions

For Males

Confidence Limits for Effective MONTH

GENDER      1         1

95% Confidence Limits

Prob         MONTH         Lower         Upper

.01     -11.21390     -49.97444      -3.05681

.02      -8.88918     -43.23911      -1.63725

.03      -7.41422     -38.96799       -.73435

.04      -6.30467     -35.75649       -.05363

.05      -5.40213     -33.14536        .50126

.06      -4.63393     -30.92388        .97455

.07      -3.96037     -28.97696       1.39042

.08      -3.35727     -27.23454       1.76359

.09      -2.80878     -25.65064       2.10375

.10      -2.30390     -24.19340       2.41761

.15       -.21354     -18.17002       3.72702

.20       1.44782     -13.40033       4.78521

.25       2.87311      -9.33024       5.71491

.30       4.15307      -5.70589       6.58054

.35       5.33914      -2.39556       7.43084

.40       6.46461        .66083       8.32248

.45       7.55351       3.45078       9.35229

.50       8.62515       5.85315      10.70912

.55       9.69678       7.68676      12.63471

.60      10.78569       8.99479      15.14645

.65      11.91115      10.01863      18.07065

.70      13.09722      10.93403      21.31588

.75      14.37718      11.83491      24.90497

.80      15.80247      12.78553      28.95414

.85      17.46383      13.85715      33.71040

.90      19.55419      15.17585      39.72450

.91      20.05908      15.49124      41.18020

.92      20.60757      15.83286      42.76263

.93      21.21066      16.20743      44.50366

.94      21.88422      16.62464      46.44924

.95      22.65242      17.09924      48.66941

.96      23.55496      17.65542      51.27925

.97      24.66452      18.33744      54.48944

.98      26.13947      19.24171      58.75919

.99      28.46419      20.66285      65.49295

Table of Probit Model Predictions (cont.)

For Females

Confidence Limits for Effective MONTH

GENDER      2         2

95% Confidence Limits

Prob         MONTH         Lower         Upper

.01     -16.26550     -64.24189      -6.19771

.02     -13.94078     -57.50252      -4.78220

.03     -12.46582     -53.22787      -3.88282

.04     -11.35626     -50.01304      -3.20544

.05     -10.45373     -47.39862      -2.65384

.06      -9.68553     -45.17383      -2.18386

.07      -9.01197     -43.22355      -1.77135

.08      -8.40887     -41.47767      -1.40163

.09      -7.86038     -39.89020      -1.06504

.10      -7.35550     -38.42925       -.75491

.15      -5.26513     -32.38438        .53303

.20      -3.60378     -27.58605       1.56257

.25      -2.17848     -23.47560       2.45192

.30       -.89853     -19.79114       3.25743

.35        .28754     -16.38519       4.01211

.40       1.41301     -13.16393       4.73888

.45       2.50191     -10.06199       5.45670

.50       3.57355      -7.03097       6.18489

.55       4.64519      -4.03520       6.94832

.60       5.73409      -1.05482       7.78771

.65       6.85955       1.89443       8.78649

.70       8.04563       4.69772      10.14383

.75       9.32558       7.06270      12.26882

.80      10.75088       8.82453      15.50679

.85      12.41223      10.27770      19.88150

.90      14.50259      11.78928      25.70271

.91      15.00748      12.13165      27.13144

.92      15.55597      12.49769      28.68944

.93      16.15907      12.89456      30.40817

.94      16.83263      13.33230      32.33323

.95      17.60083      13.82601      34.53429

.96      18.50336      14.40025      37.12606

.97      19.61292      15.09973      40.31880

.98      21.08788      16.02155      44.57100

.99      23.41260      17.46195      51.28549

Model 7

Multivariate Probit Models with Multiple Groups and Two Predictor Variables

Groups = males & females

Predictors = month & age

Probit male = a + b1 (month) + b2 (age)

Probit female = a + b1 (month) + b2 (age)

Probit male = 7.17239 + 0.13603 (month) – 0.59618 (age)

Probit female = 7.83981 + 0.13603 (month) – 0.59618 (age)

Results of the Probit Analysis

Parameter estimates converged after 17 iterations.

Optimal solution found.

Parameter Estimates (PROBIT model:  (PROBIT(p)) = Intercept + BX):

Regression Coeff.  Standard Error     Coeff./S.E.

MONTH              .13603          .04103         3.31503

AGE               -.59618          .10452        -5.70371

Intercept  Standard Error  Intercept/S.E.  GENDER

7.17239         1.47404         4.86581         1

7.83981         1.49292         5.25132         2

Pearson  Goodness-of-Fit  Chi Square =     12.938    DF = 38   P = 1.000

Parallelism Test Chi Square =       .139    DF = 1   P =  .709

Since Goodness-of-Fit Chi square is NOT significant, no heterogeneity

factor is used in the calculation of confidence limits.

The paraellelism chi-square test found no significant differences between the group slopes for month and age and computed a common slope for each variable for each group.

Probit male = 7.17239 + 0.13603 (month) – 0.59618 (age)

Probit female = 7.83981 + 0.13603 (month) – 0.59618 (age)

The groups differ in the intercept; males have a lower likelihood of success regardless of months of supervision or age.

Residual Analysis

Observed and Expected Frequencies

Number of    Observed    Expected

GENDER        MONTH   Subjects   Responses   Responses    Residual     Prob

1          6.00        7.0         3.0       4.159      -1.159   .59417

1          6.00        6.0         2.0       2.161       -.161   .36021

1          6.00        6.0         2.0       1.020        .980   .17002

1          7.00        7.0         4.0       4.521       -.521   .64591

1          7.00        7.0         2.0       2.885       -.885   .41221

1          7.00        7.0         2.0       1.447        .553   .20666

1          8.00        6.0         5.0       4.171        .829   .69509

1          8.00        6.0         3.0       2.795        .205   .46580

1          8.00        6.0         1.0       1.486       -.486   .24761

1          9.00        5.0         4.0       3.705        .295   .74098

1          9.00        5.0         3.0       2.600        .400   .52002

1          9.00        5.0         2.0       1.463        .537   .29254

1         10.00        8.0         6.0       6.264       -.264   .78301

1         10.00        8.0         5.0       4.591        .409   .57386

1         10.00        8.0         2.0       2.727       -.727   .34092

1         11.00        8.0         6.0       6.566       -.566   .82080

1         11.00        8.0         5.0       5.011       -.011   .62637

1         11.00        7.0         3.0       2.745        .255   .39207

1         12.00        6.0         5.0       5.125       -.125   .85416

1         12.00        6.0         4.0       4.060       -.060   .67663

1         12.00        8.0         4.0       3.561        .439   .44516

2          6.00        7.0         5.0       5.722       -.722   .81745

2          6.00        6.0         4.0       3.729        .271   .62154

2          6.00        6.0         2.0       2.323       -.323   .38719

2          7.00        7.0         7.0       5.959       1.041   .85123

2          7.00        7.0         5.0       4.704        .296   .67204

2          7.00        7.0         3.0       3.081       -.081   .44014

2          8.00        6.0         6.0       5.283        .717   .88055

2          8.00        6.0         5.0       4.317        .683   .71958

2          8.00        6.0         2.0       2.965       -.965   .49418

2          9.00        5.0         5.0       4.528        .472   .90554

2          9.00        5.0         3.0       3.818       -.818   .76350

2          9.00        5.0         2.0       2.742       -.742   .54833

2         10.00        8.0         8.0       7.412        .588   .92645

2         10.00        8.0         6.0       6.427       -.427   .80335

2         10.00        8.0         5.0       4.813        .187   .60159

2         11.00        8.0         7.0       7.549       -.549   .94361

2         11.00        8.0         6.0       6.711       -.711   .83883

2         11.00        7.0         5.0       4.571        .429   .65302

2         12.00        6.0         6.0       5.745        .255   .95745

2         12.00        6.0         5.0       5.219       -.219   .86986

2         12.00        8.0         6.0       5.614        .386   .70178

Predictions from Probit Equations for 14 Year Old Male & Female Probationers with 8 Months of Supervision

Male Probationers

Total = 6, 3 successful or 50%

Probit = 7.17239 + 0.13603 (8 months) –

0.59618 (14 years old)

Probit = -0.08589

Z of -0.08589 has an associated area under

the smaller portion of the curve of 0.4658

(6 juveniles) (0.4658) = 2.795 successes

Female Probationers

Total = 6, 5 successful or 83.3%

Probit = 7.83981 + 0.13603 (8 months) –

0.59618 (14 years old)

Probit = +0.58153

Z of +0.58153 has an associated area under

the larger portion of the curve of 0.71958

(6 juveniles) (0.71958) = 4.317 successes

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