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1 Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: Data Analysis Brief Book (glossary) Exploratory Data Analysis Statistical Data Analysis Using Excel for Data Analysis (using Excel) 2 Copyright 2003 John Wiley & Sons, Inc. Sekaran/RESEARCH 4E 3 FIGURE 12.1 Data Analysis Get the feel for the data Get Mean, variance' and standard deviation on each variable See if for all items, responses range all over the scale, and not restricted to one end of the scale alone. Obtain Pearson Correlation among the variables under study. Get Frequency Distribution for all the variables. Tabulate your data. Describe your sample's key characteristics (Demographic details of sex composition, education, age, length of service, etc. ) See Histograms, Frequency Polygons, etc. 4 Quantitative Data Descriptive Statistics Describing key features of data Each type of data requires different analysis method(s): Nominal Labeling No inherent value basis Categorization purposes only Ordinal Ranking, sequence Interval Relationship basis (e.g. age) Central Tendency Mean, median mode Spread Variance, standard deviation, range Distribution (Shape ) Skewness, kurtosis 5 6 1
2 Descriptive Statistics Describing key features of data Testing Goodness of Fit Nominal Identification / categorization only Ordinal (Example on pg. 139) Nonparametric statistics Do not assume equal intervals Frequency counts Averages (median and mode) Interval Parametric Mean, Standard Deviation, variance 7 Reliability Validity Involves and Factor Analysis Split Half Internal Consistency Convergent Discriminant Factorial 8 Testing Hypotheses Use appropriate statistical analysis Ttest (single or twintailed) Test the significance of differences of the mean of two groups ANOVA Test the significance of differences among the means of more than two different groups, using the F test. Regression (simple or multiple) Establish the variance explained in the DV by the variance in the Ivs catterp.htm 9 Statistical Power Claiming a significant difference Errors in Methodology Type 1 error Reject the null hypothesis when you should not. Called an alpha error Type 2 error Fail to reject the null hypothesis when you should. Called a beta error Statistical power refers to the ability to detect true differences avoiding type 2 errors 10 Statistical Power see discussion at Depends on 4 issues Sample size The effect size you want to detect The alpha (type 1 error rate) you specify The variability of the sample Too little power Overlook effect Too much power Any difference is significant Parametric vs. nonparametric Parametric (characteristics referring to specific population parameters) Parametric assumptions Independent samples Homogeneity of variance Data normally distributed Interval or better scale Nonparametric assumptions Sometimes independence of samples
3 ttests (Look at t tables; p. 435) Used to compare two means or one observed mean against a guess about a hypothesized mean For large samples t and z can be considered equivalent Calculate t =  µ S Where S is the standard error of the mean, S/ n and df = n1 13 ttests Statistical programs will give you a choice between a matched pair and an independent ttest. Your sample and research design determine which you will use. 14 ztest for Proportions (Look at t tables; p. 435) When data are nominal Describe by counting occurrences of each value From counts, calculate proportions Compare proportion of occurrence in sample to proportion of occurrence in population Hypotheses testing allows only one of two outcomes: success or failure ztest for Proportions (Look at t tables; p. 435) Comparing sample proportion to the population proportion H 0 : π = k, where k is a value between 0 and 1 H 1 : π k z = p π = p π σ p (π(1 π)/n) Equivalent to χ 2 for df = ChiSquare Test(sampling distribution) One Sample Measures sample variance Squared deviations from the mean based on normal distribution Nonparametric Compare expected with observed proportion H 0 : Observed proportion = expected proportion df = number of data points categories, cells (k) minus 1 Univariate z Test Test a guess about a proportion against an observed sample; eg., MBAs constitute 35% of the managerial population H 0 : π =.35 H 1 : π 5.35 (twotailed test suggested) χ 2 = (O E)2 E
4 Univariate Tests Some univariate tests are different in that they are among statistical procedures where you, the researcher, set the null hypothesis. In many other statistical tests the null hypothesis is implied by the test itself. 19 Contingency Tables Relationship between nominal variables Relationship between subjects' scores on two qualitative or categorical variables (Early childhood intervention) If the columns are not contingent on the rows, then the rows and column frequencies are independent. The test of whether the columns are contingent on the rows is called the chi square test of independence. The null hypothesis is that there is no relationship between row and column frequencies. 20 A statistical summary of the degree and direction of association between two variables Correlation itself does not distinguish between independent and dependent variables Most common Pearson s r You believe that a linear relationship exists between two variables The range is from 1 to +1 R 2, the coefficient of determination, is the % of variance explained in each variable by the other r = S xy /S x S y or the covariance between x and y divided by their standard deviations Calculations needed The means, xbar and ybar Deviations from the means, (x xbar) and (y ybar) for each case The squares of the deviations from the means for each case to insure positive distance measures when added, (x  x bar) 2 and (y ybar) 2 The cross product for each case (x x bar) times (y ybar) 23 The null hypothesis for correlations is H 0 : ρ = 0 and the alternative is usually H 1 : ρ 0 However, if you can justify it prior to analyzing the data you might also use H 1 : ρ > 0 or H 1 : ρ < 0, a onetailed test 24 4
5 Alternative measures Spearman rank correlation, r ranks r ranks and r are nearly always equivalent measures for the same data (even when not the differences are trivial) Phi coefficient, r Φ, when both variables are dichotomous; again, it is equivalent to Pearson s r 25 Alternative measures Pointbiserial, r pb when correlating a dichotomous with a continuous variable If a scatterplot shows a curvilinear relationship there are two options: A data transformation, or Use the correlation ratio, η 2 (etasquared) SS within 1  SS total 26 ANOVA For two groups only the ttest and ANOVA yield the same results You must do paired comparisons when working with three or more groups to know where the means lie Multivariate Techniques Dependent variable Regression in its various forms Discriminant analysis MANOVA Classificatory or data reduction Cluster analysis Factor analysis Multidimensional scaling Linear Regression We would like to be able to predict y from x Simple linear regression with raw scores y = dependent variable s y x = independent variable s x b = regression coefficient = r xy c = a constant term The general model is y = bx + c (+e) 29 Linear Regression The statistic for assessing the overall fit of a regression model is the R 2, or the overall % of variance explained by the model R 2 = 1 = unpredictable variance total variance predictable variance total variance = 1 (s 2 e / s 2 y), where s 2 e is the variance of the error or residual 30 5
6 Linear Regression Multiple regression: more than one predictor y = b 1 x 1 + b 2 x 2 + c Each regression coefficient b is assessed independently for its statistical significance; H 0 : b = 0 So, in a statistical program s output a statistically significant b rejects the notion that the variable associated with b contributes nothing to predicting y Linear Regression Multiple regression R 2 still tells us the amount of variation in y explained by all of the predictors (x) together The Fstatistic tells us whether the model as a whole is statistically significant Several other types of regression models are available for data that do not meet the assumptions needed for leastsquares models (such as logistic regression for dichotomous dependent variables) Regression by SPSS & other Programs Methods for developing the model Stepwise: let s computer try to fit all chosen variables, leaving out those not significant and reexamining variables in the model at each step Enter: researcher specifies that all variables will be used in the model Forward, backward: begin with all (backward) or none (forward) of the variables and automatically adds or removes variables without reconsideration of variables already in the model 33 Multicollinearity Best regression model has uncorrelated IVs Model stability low with excessively correlated IVs Collinearity diagnostics identify problems, suggesting variables to be dropped High tolerance, low variance inflation factor are desirable 34 Discriminant Analysis Regression requires DV to be interval or ratio If DV categorical (nominal) can use discriminant analysis IVs should be interval or ratio scaled Key result is number of cases classified correctly MANOVA Compare means on two or more DVs (ANOVA limited to one DV) Pure MANOVA via SPSS only from command syntax Can use the general linear model though
7 Factor Analysis A data reduction technique a large set of variables can be reduced to a smaller set while retaining the information from the original data set Data must be on an interval or ratio scale E.g., a variable called socioeconomic status might be constructed from variables such as household income, educational attainment of the head of household, and average per capita income of the census block in which the person resides Cluster Analysis Cluster analysis seeks to group cases rather than variables; it too is a data reduction technique Data must be on an interval or ratio scale E.g., a marketing group might want to classify people into psychographic profiles regarding their tendencies to try or adopt new products pioneers or early adopters, early majority, late majority, laggards Factor vs. Cluster Analysis Factor analysis focuses on creating linear composites of variables Number of variables with which we must work is then reduced Technique begins with a correlation matrix to seed the process Cluster analysis focuses on cases 39 Potential Biases Asking the inappropriate or wrong research questions. Insufficient literature survey and hence inadequate theoretical model. Measurement problems Samples not being representative. Problems with data collection: researcher biases respondent biases instrument biases Data analysis biases: coding errors data punching & input errors inappropriate statistical analysis Biases (subjectivity) in interpretation of results. 40 Questions to ask: Adopted from Robert Niles Where did the data come from? How (Who) was the data reviewed, verified, or substantiated? How were the data collected? How is the data presented? What is the context? Cherrypicking? Be skeptical when dealing with comparisons Spurious correlations Copyright 2003 John Wiley & Sons, Inc. Sekaran/RESEARCH 4E 41 FIGURE
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