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An Illustrative Example of Logistic Regression

This is the sample problem presented in the text

on pages 314 to 321. Consistent with the

authors strategy for presenting the problem, we

will divide the data set into a learning sample

and a validation sample, after a brief overview

of logistic regression.

An Illustrative Example of Logistic Regression

Overview of Logistic Regression - 1

Multiple regression requires that the dependent

variable be a metric variable. There are,

however, many problems in which the dependent

variable is a non-metric class or category and

the goal of our analysis is to produce a model

that predicts group membership or classification.

For example, we might be interested in predicting

whether individuals will succeed or fail in some

treatment, i.e. the likelihood that they will be

a member of a particular outcome group. We will

look at two strategies for addressing this type

of problem discriminant analysis and logistic

regression. Discriminant analysis can be used for

any number of groups. Logistic regression is

commonly used with two groups, i.e. the dependent

variable is dichotomous. Discriminant analysis

requires that our data meet the assumptions of

multivariate normality and equality of

variance-covariance across groups. Logistic

regression does not require these assumptions

"In logistic regression, the predictors do not

have to be normally distributed, linearly

related, or of equal variance within each group."

(Tabachnick and Fidell, page 575) Logistic

regression predicts the probability that the

dependent variable event will occur given a

subject's scores on the independent variables.

The predicted values of the dependent variable

can range from 0 to 1. If the probability for an

individual case is equal to or above some

threshold, typically 0.50, then our prediction is

that the event will occur. Similarly, if the

probability for an individual case is less than

0.50, then our prediction is that the event will

not occur. One of the criticisms of logistic

regression is that its group prediction does not

take into account the relative position of a case

within the distribution, a case that has a

probability of .51 is classified in the same

group as a case that has a probability of .99,

since both are above the .50 cutoff. The

dependent variable plotted against the

independent variables follows an s-shaped curve,

like that shown in the text on page 277. The

relationship between the dependent and

independent variable is not linear.

An Illustrative Example of Logistic Regression

Overview of Logistic Regression - 2

As with multiple regression, we are concerned

about the overall fit, or strength of the

relationship between the dependent variable and

the independent variables, but the statistical

measures of the fit are different than those

employed in multiple regression. Instead of the

F-test for overall significance of the

relationship, we will interpret the Model

Chi-Square statistic which is the test of a model

which has no independent variables versus a model

that has independent variables. There is a

"pseudo R square" statistic that can be computed

and interpreted as an R square value. We can also

examine a classification table of predicted

versus actual group membership and use the

accuracy of this table in evaluating the utility

of the statistical model. The coefficients for

the predictor variables measure the change in the

probability of the occurrence of the dependent

variable event in log units. Since the B

coefficients are in log units, we cannot directly

interpret their meaning as a measure of change in

the dependent variable. However, when the B

coefficient is used as a power to which the

natural log (2.71828) is raised, the result

represents an odds ratio, or the probability that

an event will occur divided by the probability

that the event will not occur. If a coefficient

is positive, its transformed log value will be

greater than one, meaning that the event is more

likely to occur. If a coefficient is negative,

its transformed log value will be less than one,

and the odds of the event occurring decrease. A

coefficient of zero (0) has a transformed log

value of 1.0, meaning that this coefficient does

not change the odds of the event one way or the

other. We can state the information in the odds

ratio for dichotomous independent variables as

subjects having or being the independent variable

are more likely to have or be the dependent

variable, assuming the that a code of 1

represents the presence both the independent and

the dependent variable. For metric independent

variables, we can state that subjects having more

of the independent variable are more likely to

have or be the dependent variable, assuming that

the independent variable and the dependent

variable are both coded in this direction.

An Illustrative Example of Logistic Regression

Overview of Logistic Regression - 3

There are several numerical problems that can

occur in logistic regression that are not

detected by SPSS or other statistical packages

multicollinearity among the independent

variables, zero cells for a dummy-coded

independent variable because all of the subjects

have the same value for the variable, and

"complete separation" whereby the two groups in

the dependent event variable can be perfectly

separated by scores on one or a combination of

the independent variables. All of these problems

produce large standard errors (I recall the

threshold as being over 2.0, but I am unable to

find a reference for this number) for the

variables included in the analysis and very often

produce very large B coefficients as well. If we

encounter large standard errors for the predictor

variables, we should examine frequency tables,

one-way ANOVAs, and correlations for the

variables involved to try to identify the source

of the problem. Like multiple regression and

discriminant analysis, we are concerned with

detecting outliers and influential cases and the

effect that they may be having on the model.

Finally, we can use diagnostic plots to evaluate

the fit of the model to the data and to identify

strategies for improving the relationship

expressed in the model. Sample size, power, and

the ratio of cases to variables are important

issues in logistic regression, though the

specific information is less readily available.

In the absence of any additional information, we

may employ the standards required for multiple

regression.

An Illustrative Example of Logistic Regression

Preliminary Division of the Data Set

The data for this problem is the Hatco.Sav data

set. Instead of conducting the analysis with the

entire data set, and then splitting the data for

the validation analysis, the authors opt to

divide the sample prior to doing the analysis.

They use the estimation or learning sample of 60

cases to build the discriminant model and the

other 40 cases for a holdout sample to validate

the model. To replicate the author's analysis,

we will create a randomly generated variable,

randz, to split the sample. We will use the

cases where randz 0 to create the logistic

regression model and apply that model to the

validation sample to estimate the model's true

accuracy rate. Note the results produced in the

chapter example were obtained by using the same

random seed and compute statement as the

two-group discriminant analysis, not the SPSS

syntax commands specified in the text on page 707.

An Illustrative Example of Logistic Regression

Specify the Random Number Seed

An Illustrative Example of Logistic Regression

Compute the random selection variable

An Illustrative Example of Logistic Regression

Stage One Define the Research Problem

- In this stage, the following issues are

addressed - Relationship to be analyzed
- Specifying the dependent and independent

variables - Method for including independent variables

Relationship to be analyzed

The research problem is still to determine if

differences in perception of HATCO can

distinguish between customers using specification

buying versus total value analysis (text, page

314).

An Illustrative Example of Logistic Regression

Method for including independent variables

Specifying the dependent and independent variables

- The dependent variable is
- X11 'Purchasing Approach',
- a dichotomous variable, where 1 indicates "Total

Value Analysis" and 0 indicates "Specification

Buying." - The independent variables are
- X1 'Delivery Speed'
- X2 'Price Level'
- X3 'Price Flexibility'
- X4 'Manufacturer Image'
- X5 'Service'
- X6 'Salesforce Image'
- X7 'Product Quality

Since the authors are interested in the best

subset of predictors, they use the forward

stepwise method for selecting independent

variables.

An Illustrative Example of Logistic Regression

Stage 2 Develop the Analysis Plan Sample Size

Issues

- In this stage, the following issues are

addressed - Missing data analysis
- Minimum sample size requirement 15-20 cases per

independent variable

Missing data analysis

There is no missing data in this problem.

Minimum sample size requirement15-20 cases per

independent variable

The data set has 60 cases and 7 independent

variables for a ratio of 9 to 1, short of the

requirement that we have 15-20 cases per

independent variable.

An Illustrative Example of Logistic Regression

Stage 2 Develop the Analysis Plan Measurement

Issues

- In this stage, the following issues are

addressed - Incorporating nonmetric data with dummy variables
- Representing Curvilinear Effects with Polynomials
- Representing Interaction or Moderator Effects

Incorporating Nonmetric Data with Dummy Variables

All of the independent variables are metric.

Representing Curvilinear Effects with Polynomials

We do not have any evidence of curvilinear

effects at this point in the analysis.

Representing Interaction or Moderator Effects

We do not have any evidence at this point in the

analysis that we should add interaction or

moderator variables.

An Illustrative Example of Logistic Regression

Stage 3 Evaluate Underlying Assumptions

- In this stage, the following issues are

addressed - Nonmetric dependent variable with two groups
- Metric or dummy-coded independent variables

Nonmetric dependent variable having two groups

The dependent variable is X11 'Purchasing

Approach', a dichotomous variable, where 1

indicates "Total Value Analysis" and 0 indicates

"Specification Buying."

Metric or dummy-coded independent variables

All of the independent variables in the analysis

are metric X1 'Delivery Speed', X2 'Price

Level', X3 'Price Flexibility', X4

'Manufacturer Image', X5 'Service', X6

'Salesforce Image', and X7 'Product Quality

An Illustrative Example of Logistic Regression

Stage 4 Estimation of Logistic Regression and

Assessing Overall Fit Model Estimation

- In this stage, the following issues are

addressed - Compute logistic regression model

Compute the logistic regression

The steps to obtain a logistic regression

analysis are detailed on the following screens.

An Illustrative Example of Logistic Regression

Requesting a Logistic Regression

An Illustrative Example of Logistic Regression

Specifying the Dependent Variable

An Illustrative Example of Logistic Regression

Specifying the Independent Variables

An Illustrative Example of Logistic Regression

Specify the method for entering variables

An Illustrative Example of Logistic Regression

Specifying Options to Include in the Output

An Illustrative Example of Logistic Regression

Specifying the New Variables to Save

An Illustrative Example of Logistic Regression

Specifying the cases to include in the analysis

An Illustrative Example of Logistic Regression

Complete the Logistic Regression Request

An Illustrative Example of Logistic Regression

Stage 4 Estimation of Logistic Regression and

Assessing Overall Fit Assessing Model Fit

- In this stage, the following issues are addressed

for the stepwise inclusion of variables - Significance test of the model log likelihood

(Change in -2LL) - Measures Analogous to R² Cox and Snell R² and

Nagelkerke R² - Classification matrices
- Check for Numerical Problems
- Once we have decided on the number of variables

to be included in the equation, we will examine

other issues of fit and compare model accuracy to

the by chance accuracy rates. - Hosmer-Lemeshow Goodness-of-fit
- By chance accuracy rates
- Presence of outliers

An Illustrative Example of Logistic Regression

Step 1 of the Stepwise Logistic Regression Model

In this section, we will examine the results

obtained at the first step of the analysis.

Initial statistics before independent variables

are included

The Initial Log Likelihood Function, (-2 Log

Likelihood or -2LL) is a statistical measure like

total sums of squares in regression. If our

independent variables have a relationship to the

dependent variable, we will improve our ability

to predict the dependent variable accurately, and

the log likelihood value will decrease. The

initial 2LL value is 78.859 on step 0, before

any variables have been added to the model.

An Illustrative Example of Logistic Regression

Significance test of the model log likelihood

At step 1, the variable X7 'Product Quality' is

added to the logistic regression equation. The

addition of this variable reduces the initial log

likelihood value (-2 Log Likelihood) of 78.859 to

37.524.

The difference between these two measures is the

model child-square value (41.335 78.859 -

37.524) that is tested for statistical

significance. This test is analogous to the

F-test for R² or change in R² value in multiple

regression which tests whether or not the

improvement in the model associated with the

additional variables is statistically

significant.

In this problem the Model Chi-Square value of

41.335 has a significance of less than 0.0001,

less than 0.05, so we conclude that there is a

significant relationship between the dependent

variable and the set of independent variables,

which includes a single variable at this step.

An Illustrative Example of Logistic Regression

Measures Analogous to R²

The next SPSS outputs indicate the strength of

the relationship between the dependent variable

and the independent variables, analogous to the

R² measures in multiple regression.

The Cox and Snell R² measure operates like R²,

with higher values indicating greater model fit.

However, this measure is limited in that it

cannot reach the maximum value of 1, so

Nagelkerke proposed a modification that does

range from 0 to 1. We will rely upon

Nagelkerke's measure as indicating the strength

of the relationship. If we applied our

interpretive criteria to the Nagelkerke R² of

0.681, we would characterize the relationship as

very strong.

An Illustrative Example of Logistic Regression

The Classification Matrices

The classification matrices in logistic

regression serve the same function as the

classification matrices in discriminant analysis,

i.e. evaluating the accuracy of the model. If

the predicted and actual group memberships are

the same, i.e. 1 and 1 or 0 and 0, then the

prediction is accurate for that case. If

predicted group membership and actual group

membership are different, the model "misses" for

that case. The overall percentage of accurate

predictions (85.00 in this case) is the measure

of the model that we rely on most heavily in

logistic regression because it has a meaning that

is readily communicated, i.e. the percentage of

cases for which our model predicts accurately.

An Illustrative Example of Logistic Regression

Correspondence of Actual and Predicted Values of

the Dependent Variable

The final measure of model fit is the Hosmer and

Lemeshow goodness-of-fit statistic, which

measures the correspondence between the actual

and predicted values of the dependent variable.

In this case, better model fit is indicated by a

smaller difference in the observed and predicted

classification. A good model fit is indicated by

a nonsignificant chi-square value.

At step 1, the Hosmer and Lemshow Test is not

statistically significant, indicating predicted

group memberships correspond closely to the

actual group memberships, indicating good model

fit.

An Illustrative Example of Logistic Regression

Check for Numerical Problems

There are several numerical problems that can

occur in logistic regression that are not

detected by SPSS or other statistical packages

multicollinearity among the independent

variables, zero cells for a dummy-coded

independent variable because all of the subjects

have the same value for the variable, and

"complete separation" whereby the two groups in

the dependent event variable can be perfectly

separated by scores on one or a combination of

the independent variables.

All of these problems produce large standard

errors (over 2) for the variables included in the

analysis and very often produce very large B

coefficients as well. If we encounter large

standard errors for the predictor variables, we

should examine frequency tables, one-way ANOVAs,

and correlations for the variables involved to

try to identify the source of the problem. Our

final step, in assessing the fit of the derived

model is to check the coefficients and standard

errors of the variables included in the model.

For the single variable included on the first

step, neither the standard error nor the B

coefficient are large enough to suggest any

problem.

An Illustrative Example of Logistic Regression

Significance test of the model log likelihood

Step 2 of the Stepwise Logistic Regression Model

In this section, we will examine the results

obtained at the second step of the analysis.

At step 2, the variable X3 'Price Flexibility' is

added to the logistic regression equation. The

addition of this variable reduces the initial log

likelihood value (-2 Log Likelihood) of 78.858931

to 20.258.

The difference between these two measures is the

model child-square value (58.601 78.859 -

20.258) that is tested for statistical

significance. This test is analogous to the

F-test for R² or change in R² value in multiple

regression which tests whether or not the

improvement in the model associated with the

additional variables is statistically

significant. In this problem the Model

Chi-Square value of 58.601 has a significance of

less than 0.0001, less than 0.05, so we conclude

that there is a significant relationship between

the dependent variable and the set of independent

variables, which now includes two independent

variables at this step.

An Illustrative Example of Logistic Regression

Measures Analogous to R²

The next SPSS outputs indicate the strength of

the relationship between the dependent variable

and the independent variables, analogous to the

R² measures in multiple regression.

If we applied our interpretive criteria to the

Nagelkerke R² of 0.852 (up from 0.681 at the

first step), we would characterize the

relationship as very strong.

An Illustrative Example of Logistic Regression

The Classification Matrices

The classification matrices in logistic

regression serve the same function as the

classification matrices in discriminant analysis,

i.e. evaluating the accuracy of the model.

The overall percentage of accurate predictions

now increases to 98.33 in this case. Only one

case is classified incorrectly.

An Illustrative Example of Logistic Regression

Correspondence of Actual and Predicted Values of

the Dependent Variable

The final measure of model fit is the Hosmer and

Lemeshow goodness-of-fit statistic, which

measures the correspondence between the actual

and predicted values of the dependent variable.

In this case, better model fit is indicated by a

smaller difference in the observed and predicted

classification. A good model fit is indicated by

a nonsignificant chi-square value.

At step 2, the Hosmer and Lemshow Test is not

statistically significant, indicating predicted

group memberships correspond closely to the

actual group memberships, indicating good model

fit.

An Illustrative Example of Logistic Regression

Check for Numerical Problems

Our check for numerical problems is a check for

standard errors larger than 2 or unusually large

B coefficients.

We do not identify any problems from the table of

variables in the equation.

An Illustrative Example of Logistic Regression

Significance test of the model log likelihood

Step 3 of the Stepwise Logistic Regression Model

In this section, we will examine the results

obtained at the third step of the analysis.

At step 3, the variable X5 'Service' is added to

the logistic regression equation. The addition

of this variable reduces the initial log

likelihood value (-2 Log Likelihood) of 78.859 to

6.254.

The difference between these two measures is the

model child-square value (72.605 78.859 -

6.254) that is tested for statistical

significance. This test is analogous to the

F-test for R² or change in R² value in multiple

regression which tests whether or not the

improvement in the model associated with the

additional variables is statistically

significant. In this problem the Model

Chi-Square value of 72.605 has a significance of

less than 0.0001, less than 0.05, so we conclude

that there is a significant relationship between

the dependent variable and the set of independent

variables, which now includes three independent

variables at this step.

An Illustrative Example of Logistic Regression

Measures Analogous to R²

The next SPSS outputs indicate the strength of

the relationship between the dependent variable

and the independent variables, analogous to the

R² measures in multiple regression.

If we applied our interpretive criteria to the

Nagelkerke R² of 0.960 (up from 0.852 at the

previous step), we would characterize the

relationship as very strong.

An Illustrative Example of Logistic Regression

Correspondence of Actual and Predicted Values of

the Dependent Variable

The final measure of model fit is the Hosmer and

Lemeshow goodness-of-fit statistic, which

measures the correspondence between the actual

and predicted values of the dependent variable.

In this case, better model fit is indicated by a

smaller difference in the observed and predicted

classification. A good model fit is indicated by

a nonsignificant chi-square value.

At step 3, the Hosmer and Lemshow Test is not

statistically significant, indicating predicted

group memberships correspond closely to the

actual group memberships, indicating good model

fit.

An Illustrative Example of Logistic Regression

The Classification Matrices

The classification matrices in logistic

regression serve the same function as the

classification matrices in discriminant analysis,

i.e. evaluating the accuracy of the model.

The overall percentage of accurate predictions

for the three variable model is 98.33. Only two

cases are classified incorrectly.

An Illustrative Example of Logistic Regression

Check for Numerical Problems

Our check for numerical problems is a check for

standard errors larger than 2 or unusually large

B coefficients.

The standard errors for all of the variables in

the model are substantially larger than 2,

indicating a serious numerical problem. In

addition, the B coefficients have become very

large (remember that these are log values, so the

corresponding decimal value would appear much

larger). This model should not be used, and we

should interpret the model obtained at the

previous step. In hindsight, we may have gotten

a notion that a problem would occur in this step

from the classification table at the previous

step. Recall that we had only one

misclassification on the previous step, so there

was almost no overlap remaining between the

groups of the dependent variable.

An Illustrative Example of Logistic Regression

Returning to the two-variable model

The residual and Cook's distance measures which

we have available are for the three variable

model which SPSS was working with at the time it

concluded the stepwise selection of variables.

Since I do not know of a way to force SPSS to

stop at step 2, I will repeat the analysis using

direct entry for the two independent variables

which were found to be significant with stepwise

selection X7 'Product Quality' and X3 'Price

Flexibility.'

An Illustrative Example of Logistic Regression

Re-run the Logistic Regression

An Illustrative Example of Logistic Regression

Complete the specification for the new analysis

An Illustrative Example of Logistic Regression

The Two-Variable Model

To sum up evidence of model fit presented

previously, the Model Chi-Square value of 58.601

has a significance of less than 0.0001, less than

0.05, so we conclude that there is a significant

relationship between the dependent variable and

the two independent variables. The Nagelkerke R²

of 0.852 would indicate that the relationship is

very strong. The Hosmer and Lemeshow

goodness-of-fit measure has a value of 10.334

which has the desirable outcome of

nonsignificance.

An Illustrative Example of Logistic Regression

The Classification Matrices

The classification matrices in logistic

regression serve the same function as the

classification matrices in discriminant analysis,

i.e. evaluating the accuracy of the model.

The overall percentage of accurate predictions

(98.33 in this case) is very high, with only one

case being misclassified. To evaluate the

accuracy of the model, we compute the

proportional by chance accuracy rate and the

maximum by chance accuracy rates, if appropriate.

The proportional by chance accuracy rate is

equal to 0.536 (0.6332 0.3672). A 25

increase over the proportional by chance accuracy

rate would equal 0.669. Our model accuracy race

of 98.3 exceeds this criterion. Since one of

our groups contains 63.3 of the cases, we might

also apply the maximum by chance criterion. A

25 increase over the largest groups would equal

0.792. Our model accuracy race of 98.3 also

exceeds this criterion. In addition, the

accuracy rates for the unselected validation

sample, 87.50, surpasses both the proportional

by chance accuracy rate and the maximum by chance

accuracy rate.

An Illustrative Example of Logistic Regression

The Classification Matrices

SPSS provides a visual image of the

classification accuracy in the stacked histogram

as shown below. To the extent to which the

cases in one group cluster on the left and the

other group clusters on the right, the predictive

accuracy of the model will be higher.

An Illustrative Example of Logistic Regression

Presence of outliers

There are two outputs to alert us to outliers

that we might consider excluding from the

analysis listing of residuals and saving Cook's

distance scores to the data set. SPSS provides

a casewise list of residuals that identify cases

whose residual is above or below a certain number

of standard deviation units. Like multiple

regression there are a variety of ways to compute

the residual. In logistic regression, the

residual is the difference between the observed

probability of the dependent variable event and

the predicted probability based on the model.

The standardized residual is the residual divided

by an estimate of its standard deviation. The

deviance is calculated by taking the square root

of -2 x the log of the predicted probability for

the observed group and attaching a negative sign

if the event did not occur for that case. Large

values for deviance indicate that the model does

not fit the case well. The studentized residual

for a case is the change in the model deviance if

the case is excluded. Discrepancies between the

deviance and the studentized residual may

identify unusual cases. (See the SPSS chapter on

Logistic Regression Analysis for additional

details). In the output for our problem, SPSS

listed one case that may be considered an outlier

with a studentized residuals greater than 2, case

13

An Illustrative Example of Logistic Regression

Cooks Distance

SPSS has an option to compute Cook's distance as

a measure of influential cases and add the score

to the data editor. I am not aware of a precise

formula for determining what cutoff value should

be used, so we will rely on the more traditional

method for interpreting Cook's distance which is

to identify cases that either have a score of 1.0

or higher, or cases which have a Cook's distance

substantially different from the other. The

traditional method for detecting unusually large

Cook's distance scores is to create a scatterplot

of Cook's distance scores versus case id or case

number.

An Illustrative Example of Logistic Regression

Request the Scatterplot of Cook's Distances

An Illustrative Example of Logistic Regression

Specifying the Variables for the Scatterplot

An Illustrative Example of Logistic Regression

The Scatterplot of Cook's Distances

On the plot of Cook's distances, we see a case

that exceeds the 1.0 rule of thumb for

influential cases and has a distance value much

different than the other cases. This is actually

the same case that was identified as an outlier

on the casewise plot, though it is difficult to

track down because SPSS uses the case number in

the learning sample for the casewise plot. This

case is the only case in the two variable model

that was misclassified. We cannot omit it

because we would again be faced with no overlap

between the groups, producing the problematic

numeric results that we found with the three

variable model.

An Illustrative Example of Logistic Regression

Stage 5 Interpret the Results

- In this section, we address the following issues
- Identifying the statistically significant

predictor variables - Direction of relationship and contribution to

dependent variable

Identifying the statistically significant

predictor variables

The coefficients are found in the column labeled

B, and the test that the coefficient is not zero,

i.e. changes the odds of the dependent variable

event is tested with the Wald statistic, instead

of the t-test as was done for the individual B

coefficients in the multiple regression equation.

The Wald tests for the two independent variables

X7 'Product Quality' and X3 'Price Flexibility'

are both statistically significant (p lt 0.05), as

we knew they would be from the first two steps of

the stepwise procedure.

An Illustrative Example of Logistic Regression

Direction of relationship and contribution to

dependent variable

The negative sign of X7 'Product Quality'

indicates an inverse relationship with the

dependent variable. As the rating for X7

'Product Quality' increases, there is a higher

likelihood that the respondent favored the

'Specification Buying' approach to purchasing.

In contrast the positive coefficient for X3

'Price Flexibility' indicates a direct

relationship to the dependent variable. As the

rating for X3 'Price Flexibility', there is a

higher likelihood that the respondent favored the

'Total Value Analysis' approach to purchasing.

Since the B coefficient is expressed in log

units, we cannot directly interpret the magnitude

of the change associated with a one unit change

in the independent variable. However, if we

convert the B coefficient from log to decimal

units, the result represents the change in the

odds of having the dependent variable event for a

one unit of change in the independent variable.

For a one-unit change in X3 'Price Flexibility',

the odds of being in the 'Total Value Analysis'

approach to purchasing increase by 6 times. A

one unit change in X7 'Product Quality' reduces

the odds of being in the 'Total Value Analysis'

approach to purchasing by 1/20th (i.e. you have a

much lower probability of being in the 'Total

Value Analysis' approach to purchasing). Should

we want to talk about the odds ratio for a

different level of change, e.g. a 2-unit change

in the independent variable, we cannot simply

multiply the odds-ratio for a one unit change by

2. Instead we would multiply the B coefficient

by 2 and take the antilog EXP(B) of that value.

For example, in this problem, EXP(1x1.8035)

6.2371, while the odds ratio for a two unit

change in X3 would be EXP(2x1.8035) 38.900.

An Illustrative Example of Logistic Regression

Stage 6 Validate the Model

By holding out 40 of the subjects as a

validation sample, the authors created the

context for validation analysis. As stated

above, the accuracy rate for the validation

sample (87.50) is in the same range as the

accuracy rate for the learning sample (98.33).

Though the difference is larger than 10, the

very high accuracy rates for both samples would

incline us to support a conclusion that the

findings are generalizable.

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You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

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Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

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