Wednesday, 31 August 2011
Discriminant Analysis - Assumptions
The underlying assumptions of Discriminant Analysis (DA) are:
– Each group is normally distributed, Discriminant Analysis is relatively robust to departures from normality.
– The groups defined by the dependent variable exist a priori.
– The Predictor variable, Xk are multivariate normally distributed, independent, and non-collinear
– The variance/covariance matrix of the predictor variable across the various groups are the same in the population, (i.e. Homogeneous)
– The relationship is linear in its parameters
– Absence of leverage point outliers
– The sample is large enough: Unequal sample sizes are acceptable. The sample size of the smallest group needs to exceed the number of predictor variables. As a “rule of thumb”, the smallest sample size should be at least 20 for a few (4 or 5) predictors. The maximum number of independent variables is n - 2, where n is the sample size. While this low sample size may work, it is not encouraged, and generally it is best to have 4 or 5 times as many observations and independent variables
– Errors are randomly distributed
Drawback of Discriminant Analysis
– An important drawback of discriminant analysis is its dependence on a relatively equal distribution of group membership. If one group within the population is substantially larger than the other group, as is often the case in real life, Discriminant analysis might classify all observations in only one group. An equal good-bad sample should be chosen for building the discriminant analysis model.
– Another significant restriction of discriminant analysis is that it can’t handle categorical independent variables.
– Discriminant analysis is more rigid than logistic regression in its assumptions. In contrast to ordinary linear regression, discriminant analysis does not have unique coefficients. Each of the coefficients depends on the other coefficients in the estimation and therefore there is no way of determining the absolute value of any coefficient.
Discriminant Analysis Vs Logistic Regression
– Similarity: Both techniques examine an entire set of interdependent relationships
Discriminant Analysis Vs ANOVA
– Similarity: Both techniques examine an entire set of interdependent relationships
– Difference: In Discriminant analysis, Independent variables are metric where as in ANOVA it is categorical.
Author: Pratik Pawar
Group: Marketing - Group 4
Discriminant analysis is done to differentiate between groups usually between 2 groups. Discriminant analysis is based on regression. The effect of independent variables is studied on dependent variables.The steps for a discriminant analysis are as follows:
1. Formulate the problem
2. Determine the discriminant function coefficients that result in the highest ratio of between-group variation to within-group.
3. Test the significance of the discriminant function.
4. Interpret the results.
5. Determine the validity of the analysis.
The steps to be followed in the SPSS software to get the relevant data are:
1. First enter the grouping variable. Then, define the lowest and highest coded value for the grouping variable by clicking on Button Define Range. Then, select the independent variables in the ‘Independents:’ box.
2. Button Statistics: Here you can indicate those statistics that are desired in discriminant analysis. Often these include means, univariate ANOVAs, unstandardized Function Coefficients.
3. Button Classify: Many classification options can be selected here, such as prior probabilities and plots. Also, a summary table can be requested.
4. Button Save: This option allows you to save as new variables: Predicted group membership, Discriminant Scores and Probabilities of group membership.
5. From the data received one of the most important value is the 'eigen value' and it is a canonical discriminant function. An eigenvalue indicates the proportion of variance explained. A large eigenvalue is associated with a strong function. The canonical relation is a correlation between the discriminant scores and the levels of the dependent variable. A high correlation indicates a function that discriminates well.
6. The Wilks Lambda is another important value. Wilks’ Lambda is the ratio of within-groups sums of squares to the total sums of squares. This is the proportion of the total variance in the discriminant scores not explained by differences among groups. A lambda of 1.00 occurs when observed group means are equal (all the variance is explained by factors other than difference between those means), while a small lambda occurs when within-groups variability is small compared to the total variability. A small lambda indicates that group means appear to differ. The associated significance value indicates whether the difference is significant.
7. ‘Functions at Group Centroids’ indicates the average discriminant score for subjects in the two groups. More specifically, the discriminant score for each group when the variable means (rather than individual values for each subject) are entered into the discriminant equation.
8. The ‘Canonical Discriminant Function Coefficients’ indicate the unstandardized scores concerning the independent variables. It is the list of coefficients of the unstandardized discriminant equation. Each subject’s discriminant score would be computed by entering his or her variable values (raw data) for each of the variables in the equation.
Author: Gayatri Nair
Group: Marketing - Group 4
Applications of Discriminant Analysis
Discriminant analysis is a statistical technique widely used in the business world. Discriminant analysis uses a collection of interval variables to predict a categorical variable that may be a dichotomy or have more than two values.
The technique involves finding a linear combination of independent variables (predictors) – the discriminant function – that creates the maximum difference between group membership in the categorical dependent variable.
Thus, DA is used when:
· the dependent variable is categorical with the predictor independent variables interval level such as age, income, attitudes, perceptions, and years of education.
· there are more than two Dependant Variable categories, unlike logistic regression, which is limited to a dichotomous dependent variable.
Discriminant analysis is used to forecast the outcome of a variety of variables that impact the profitability of a business. Classic examples of the applicantion of discriminant analysis include:
- Performing an default risk evaluation of loan applicants;
- Benchmarking of potential job applicants;
- Forecasting insurance risk
- Predicting academic performance from historical data
- Developing auditing patterns
- Fraud management
Discriminant analysis is most often used to help researchers analyze the group or category to which a subject belongs. Let us look at two examples.
Judging the credit worthiness of a loan-applicant
Discriminant analysis has been used with success in consumer credit and other forms of instalment lending in which various characteristics of an individual are quantitatively rated and a credit decision is made on the basis of the total score. The plastic credit cards many of us carry often are given out on the basis of a credit scoring system that takes into account such things as age, occupation, duration of employment, home ownership, years of residence, telephone, and annual income.
Numerical rating systems also are used by companies extending trade credit. With the overall growth of trade credit, a number of companies are finding it worthwhile to screen out "clear" accept and reject applicants. In other words, routine credit decisions are made on the basis of a numerical score.
Marginal applicants, who fall between "clear" accept or reject signals, can then be analyzed in detail by the credit analyst. In this way, a company is able to achieve greater efficiency in its credit investigation process. It uses trained credit analysts to the best advantage.
Judging the suitability of a candidate for a job.
When individuals are interviewed for a job, managers do not know for sure how job candidates will perform on the job if hired. Suppose, however, that a human resource manager has a list of current employees who have been classified into two groups: "high performers" and "low performers." These individuals have been working for the company for some time, have been evaluated by their supervisors, and are known to fall into one of these two mutually exclusive categories.
The manager also has information on the employees' backgrounds: educational attainment, prior work experience, participation in training programs, work attitude measures, personality characteristics, and so forth. This information was known at the time these employees were hired. The manager wants to be able to predict, with some confidence, which future job candidates are high performers and which are not.
A researcher or consultant can use discriminant analysis, along with existing data, to help in this task.
There are two basic steps in discriminant analysis. The first involves estimating coefficients, or weighting factors, that can be applied to the known characteristics of job candidates (i.e., the independent variables) to calculate some measure of their tendency or propensity to become high performers. This measure is called a "discriminant function." Second, this information can then be used to develop a decision rule that specifies some cut-off value for predicting which job candidates are likely to become high performers.
The tendency of an individual to become a high performer can be written as a linear equation. The values of the various predictors of high performer status (i.e., independent variables) are multiplied by "discriminant function coefficients" and these products are added together to obtain a predicted discriminant function score.
This score is used in the second step to predict the job candidates likelihood of becoming a high performer.
There are more complicated cases, in which the dependent variable has more than two categories. Discriminant analysis allows for such a case, as well as many more categories and this is where it scores over multivariate regression analysis. The interpretation, however, of the discriminant function scores and coefficients becomes more complex
Group: Finance 3
In today’s class we were introduced to the concept of Discriminant analysis. The main purpose of a discriminant function analysis is to predict group membership based on a linear combination of the interval variables.
In simple words, the analysis helps into finding which item or object belongs to a particular group or classification based on certain characteristics. It differs from group building techniques such as cluster analysis in that the classifications or groups to choose from must be known in advance.
To take an example, let us assume that we have data on 80 students in Business Analytics Class. We have data on number of students who want a job in Analytics and Data Modelling and number of students who want in Sales and Distribution. We need to predict group membership by looking at independent variables which may include: Students with engineering background, age, gender, number of work experience years.
The discriminant function analysis thus helps to predict group membership when only independent variables are known. It shows the relationship between the dependent variable (Students interested in analytics job and students interest in Sales job) and interval variables (Age, gender, number of work experience years, education background etc). The analysis shows that students with engineering background and students with more work experience wanted job in Analytics industry. Also, the students wanting job in sales showed a trend of students who did not have work experience, younger age (22-24 years) and were males.
Thus, this model helps to predict membership for group of students wanting job in Analytics and Sales based on observed variables.
In class, the concept was explained to us with the help of an example “Bank Loan” where the dependent variable was “previously defaulted” and independent variable was (age, level of education, years at current address etc). We did a regression analysis to find out whether an individual will default or not. A score was computed which was the base to our conclusion whether an individual will default or not.
Another scenario where Discriminant Analysis can be used is to find the variable which predicts the number of students using cell phones in class and students not using. The observed variables can b: Age, academic grades, member of committee, a persona having girlfriend/ boyfriend, etc. Thus, we can find characteristics essential to classify students into two groups: Students using cell phones and students not using cell phones. The relationship established can be used to predict the groups and have proper mix in all classes and avoid sending students of similar group in the same class.
This analysis can also be used to find: The number of customers going for insurance policy after meeting the agent of the company. The essential characteristics include: Gender, Age group, income level, education level, and awareness about insurance, etc. Thus a model can be established to understand the data set and identify characteristics affecting the decision making. Thus, the analysis can be used to assess training needs for agents meeting different class of customers to increase convertibility.
Finance - GROUP 1
Finance - GROUP 1
Today’s class was about discrimination analysis in SPSS. As per my understanding from class and some research on net, here are few things about discrimination analysis and Wilks lambda. Also I have mentions some of problems in Discriminant analysis.
To start with, discriminant analysis is done by creating a new variable that is a combination of the original predictors. This is done in such a way that the differences between the predeﬁned groups, with respect to the new variable, are maximised. Because there are two classes the maximum number of discriminant functions is one and each has an associated eigen value. Larger eigen values are associated with greater class separations and they can also be used to measure the relative importance of discriminant functions in multi-classanalyses. Eigen values can be converted into Wilk’s lambda, a multivariatetest statistic whose value ranges between zero and one. Values close to zero indicate that class means are different while values close to one indicate that the class means are not different. Wilk’s lambda will be one when the means are identical. Wilk’s lambda is equal to 1/(1+λ) for a two-class problem and can beconverted into a chi-square statistic so that a signiﬁcance test can be applied. Wilk’s lambda can be converted into a canonical correlation coefﬁcient fromthe square root of 1- Wilk’s lambda. The canonical correlation is the squareroot of the ratio of the between-groups sum of squares to the total sum of squares. Squared, it is the proportion of the total variability explained by differences between classes. Thus, if all of the variability in the predictors wasa consequence of the group differences the canonical correlation would beone, while if none of the variability was due to group differences thecanonical correlation would be zero. An assumption of a discriminant analysis is that there is no evidence of a difference between the covariance matrices of the classes. There are formalsigniﬁcance tests for this assumption (e.g. Box’s M) but they are not very robust.In particular they are generally thought to be too powerful, i.e. the nullhypothesis is rejected even when there are minor differences, and Box’s M isalso susceptible to deviations from multivariate normality (another assumption).
Each discriminant function can be summarised by three sets of coefﬁcients:
(1) standardised canonical discriminant function coefﬁcients (weights);
(2) corre-lations between discriminating variables and the standardised canonicaldiscriminant function; and
(3) unstandardised canonical discriminant functioncoefﬁcients (weights).
Weakness of Discriminant analysis
Inconsistent -. Because of the dissimilar processes involved in Fisher's discriminant analysis and Mahalanobis's discriminant analysis approaches to discriminant analysis, the resulting solutions are not alike.
Unintuitive- Due to the complexity of discriminant analysis's mechanics, it is an unwieldy tool for all those but the mathematically-savvy. Similar tools, such as multiple regression, are just as flexible as discriminant analysis without carrying along the intricacy and specificity associated with discriminant analysis.
Prediction- The prediction available through discriminant analysis is not true prediction. discriminant analysis can only tell a researcher the likely grouping of a certain data point; it cannot tell the researcher other properties of the data point or how likely it is the data point is a member of the classified group.
Inspite of all the weaknesses Discrimination analysis has wide rage of applications in all the fields.
Group- HR 1
Author- Ankita Kanojia