MARK 5343 Exam 2

Logistic Regression

Makes probabilistic predictions of whether you are a 1 or a 0

Logistic regression – what is different here regarding the DV? the IVs?

DV: Binary, dummy coded, nonmetric Y that is predicted

IV: Predictors that create a variate called Z

What are odds ratios?

What are log odds ratios?

What is the variate in logistic regression considered to be

Z

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ln(P1/P0)

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a logit

What does it mean when Z is positive

Increased P(1) (probability of event 1)

What does it mean when Z is negative

Increased P(0) (probability of event 0)

We turn the variate into P(1) and P(0)… how?

e^z over 1 + e^z = P(1)

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1 - P(1) = P(0)

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Determines strongest alignment to DV

Null model

Specifies in advance the P(1) and P(0) based on the observed DV proportions. Then, we introduce X material and hopefully improve over the null.

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ln(P1/P0) = Z + 0(X)

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Model only has the intercept and no predictors

Parameter estimation to get the best bs is Iterative

Model produced probability of being “what you are”

Likelihood function (L)

The joint probability based on the model for the entire sample (Multiplication of actual probabilities)

Maximizes L to find the best b’s

L = 1 if probabilities are perfect

Log likelihood function (LL)

Sum of the log of each probability

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Model fit diagnostic that has meatier numbers than L and should be minimized

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LL = Ln(.8) + Ln(.45) + Ln(.9)

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Can use to compute pseudo R ²

-2LL

Makes LL useable in Chi-squared test of significance

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Produces x² statistic that follows a chi-squared distribution. The closer this is to 0, the better

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We want the proposed model’s to be lower than the null’s

-2LL difference (Likelihood Ratio Test)

Difference from null model to test whether the proposed model’s estimation is significantly better than the null model

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-2(LL_null - LL_proposed)

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Can also use to compare two models

Logistic output pieces that parallel multiple regression output pieces, including if stepwise

Variate: Z vs. Y-hat

Model Fit: L, LL, pseudo R² vs. R²

b’s P-Value: Wald vs. t-statistic

Comparable X’s Impact: EXP(b) vs. Standardized coefficients (beta)

Equation: B vs. Unstandardized B

Variance Explained: pseudo R² vs. R²

Other Model Fit: F test vs. Chi-square test of -2LL difference

Pseudo R-square (use Nagelkerke)

Model fit test that determines how much variability a model can explain of its outcome (explanatory power)

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Want this number to be close to 1

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Ex. 64% of the variance in Y is explained by the predictors

Significance of individual variables in logistic regression

Wald test statistic

Sign and size of bs and the impact on P(1)

A 1 unit increase in the predictor variable results in b increase/decrease in the Y/log-odds

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Can assess directionality of relationship

Exp(b)

Can compare IVs relative impacts on P(1)

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1 = no change in the odds

exp(b) - 1 = % change in odds

What model fit test is distinct for a categorical DV

Classification Matrix

Understanding prediction accuracy in the classification matrix

correct prediction freq + correct prediction freq = % accuracy at prediction

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look at the diagonal

What are the two benchmarks for classification accuracy?

Cpro and Cmax

Cmax

The larger percentage between percent positive and percent negative in a classification matrix

Cpro

percent positive² + (percent negative)²

How to compute Cpro and Cmax criteria for judging classification (1.25 x Cpro, 1.25 x Cmax)

Cmax:

If

Cpro:

Cpro - square actual proportions and add together, the chance standard to beat by 25%

Cmax – assign everyone to largest category, percent accuracy doing that? Beat by 25%

Logistic Regression validation with a hold-out/split sample

Two group discriminant Analysis – nature of DV and IVs

Discriminant analysis big picture idea

Find best bs that make groups as separate / discriminated as possible

How many discriminant functions in relation to number of groups?

Read outputs from running in SPSS, direct (simultaneous) or stepwise

Know differences in coefficient types: unstandardized, standardized, loading/structure

Which two are most used for variable “importance”?

Discriminant analysis statistical test of overall model fit

Classification accuracy (another test of model fit/strength, higher percentages better)

How do I judge quality of classification accuracy with Cpro and Cmax? (same as logistic)

Discriminant analysis validation with a hold-out/split sample

Validation via the “one-at-a-time left out” method (labeled cross-validated in SPSS)

Three + groups discriminant analysis – nature of DV and IVs

How many discriminant functions in relation to number of groups 3+?

Tests of significance for each function

Types of coefficients for each function

Outputs for direct or stepwise estimation, SPSS, 2D plots

What is the potency index?

Use of potency index for variable “importance” – one number for each IV even if multiple functions

Classification matrix and accuracy interpretation (know what it looks like for 3+ groups)

Map with Centroids (green, blue, red, people from slides, pink x boxes are centroids)

What is conjoint, what does it allow you to do?

Ratings-based full profile method (previous textbook chapter)

Defining attributes and levels

Use of fractional factorial designs to build profiles [the design… the X side predicting ratings]

Concepts of total utility (rating) and part-worth utilities (bs)

Addition of other profiles for “validation” - hold out profiles

Addition of other profiles for simulation (see what would happen in a market with certain offerings)

Respondent task = ratings task (vs. choice task)

Effect coding (vs. dummy coding) the design matrix

Multivariate statistical technique used to estimate part-worths (regression)

The “nice” property of effects coded estimates (sum to 0)

Determine fit at individual level, statistics from the chapter (Multiple R, Tau nonparametric)

Deletion of cases with bad fit (estimation or hold out) and/or illogical part-worth patterns

Restating part worth utilities (book process – lowest level of each attribute 0)

Rescaling part worth utilities (book process – each attribute brought 100 to the table)

Computing attribute importance (High minus low for each attribute, over sum of all hi minus low, easy if low within each attribute was made 0, now just high for each attribute over the sum of all highs)

Ability to “Segment” people based on individual-level information

Use of simulation profiles to simulate market share (book uses 2 existing products, 1 new)

Get total utility for each product configuration in the simulation

Use values to compute discrete predicted choice (maximum utility rule)

Or use values to compute probabilistic predicted choice (BTL, or Logit Choice rule)

Core understanding of leap from ratings to “Choice-Based” (with HB estimation):

Choice sets

Choice alternatives

Probabilistic prediction within each choice set

HB estimation - borrowing from upper model

Individual part-worth utilities

Averaging over every nth draw from last set of draws (e.g., every 10th draw from last 1000 gives 100 values to average)

Uses of individual-level utility estimates as before

Very high level understanding of history/development: Ratings Based

Very high level understanding of history/development: Choice Based – aggregate logit model, one equation for everyone

Very high level understanding of history/development: Latent class – segments and estimates one logit model per segment (a way to represent respondent heterogeneity)

Very high level understanding of history/development: CBC-HB… individual level heterogeneity, one model (part worth utilities) per participant

Relative variable importance: Regression, Logistic, 2 group discriminant, 3 group discriminant, conjoint