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INPO Statistics Workshop: Notes and Exercises using EXCEL and SAS Developed by Jennifer Lewis Priestley, Ph.D. Kennesaw State University

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Statistics Workshop Topics MODULE ONE: Concept Review Review of Statistical Concepts Data Analysis using EXCEL MODULE TWO: Inferential testing using EXCEL and SAS Confidence Intervals Ttests ANOVA Chi-Square MODULE THREE: Predictive Modeling using EXCEL and SAS Regression Analysis Logistic Analysis Discriminant Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Review of Statistical Concepts Prior to analysis, a determination must be made of the type of variables in question. Variable type will, in many cases, dictate the analysis options. Variable types for review: Qualitative –categorical (e.g., gender, race) –ordinal (e.g., rankings, Likert data*) Quantitative –interval (e.g., temperature) –ratio (e.g., weight, height) * Mathematicians and Statisticians consider Likert data to be qualitative and therefore restrict its use to qualitative techniques such as Chi-square analysis. However, in practice, most people treat Likert data as quantitative and utilize quantitative techniques.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Descriptions of data typically include: Measurements of Central Tendency –Mean, Median, Mode ( do you know when to use each?) Measurements of Dispersion –Standard Deviation and Variance Outlier Detection: For near bell-shaped data –Use 3-Sigma (Empirical) Rule : any value that is more than 3 standard deviations above or below the mean For Skewed Data –Use Tukey’s Rule: any value that is more than one step below Q1 or more than 1 step above Q3; A step = 1.5*IQR MODULE 1: Review of Statistical Concepts

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The Central Limit Theorem forms the basis for why inferential statistics (versus descriptive statistics) is possible. Prior to reviewing the Theorem, pull up this site: http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html MODULE 1: Review of Statistical Concepts

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The distribution of sample means will, as the number of samples increases approach a normal distribution; The mean of all sample means approximates the population mean; The std of all sample means is the std of the population/the SQRT of the sample size; If the population is NOT normally distributed, sample sizes must be greater than 30 to assume normality; If the population IS normally distributed, samples can be of any size to assume normality (although greater than 30 is always preferred). MODULE 1: Review of Statistical Concepts Important concepts to remember about the Central Limit Theorem:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics A few points to remember about the Normal Distribution: Bell shaped and symmetric about the mean μ. Mean = μ, Median = μ, Mode = μ. The area under the normal curve below μ is.5. Probability that a Normal Random Variable Outcome: –Lies within +/- 1 std dev of the mean is.6826 –Lies within +/- 2 std dev of the mean is.9544 –Lies within +/- 3 std dev of the mean is.9974 For all other probabilities, convert the relevant observation to a Z-score: Z=(x- μ )/ σ Any observation that has a Z-score greater than 2 is typically considered to be a statistical outlier…since its probability of occurrence is less than 5%. MODULE 1: Review of Statistical Concepts

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Review of Statistical Concepts Ho is trueHo is false Reject Ho TYPE I ErrorValid Decision Do not reject Ho Valid DecisionTYPE II Error Hypothesis Testing:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics 1.Statement of Hypotheses —H 0 : Null Hypothesis – opposite of Alternative Hypothesis —H 1 : Alternative Hypothesis – what we are trying to prove —Evaluate Type I & Type II Errors 2.Set Significance level, —Standard of Proof, or Level of Risk —Represents Probability of Type I Error 3.Calculate the test statistic from the sample. 4.Calculate the p-value (strength of the evidence) —1. If the p-value < , Accept H1. —2. If the p-value > , Accept H0. MODULE 1: Review of Statistical Concepts

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics In this section, we will use EXCEL to execute some of the most common types of univariate/bivariate and multivariate data analysis: Descriptive Statistics Histograms Scatterplots and Charts Pivot Tables Using Formulas (fx) Look Up Tables Lagniappe MODULE 1: Data Analysis Using EXCEL

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: COMP1 Dataset Speed = Amount of time it takes to deliver a product (in days) once the order has been received. PriceLv = Perceived level of price charged by suppliers PriceFlx = Perceived willingness of COMP1 managers to negotiate price Man_Imag = Overall image of manufacturer Service = Overall level of service necessary to maintain a satisfactory relationship with customer Sal_Imag = Overall perceived image of salesforce Quality = Perceived level of product quality The dataset used throughout this workshop is the COMP1 dataset, with 14 variables and 100 observations. Size = Large (1) or Small (0) Usage = Percentage of total product purchased from COMP1 Satisf = How satisfied purchaser is with Comp1 SpecBuy = Extent to which a purchaser evaluates each purchase separately (1=each purchase evaluated separately, 0 = lot buying) Procure = Centralization of purchase decisions (1= Centralized, 0=decentralized) Ind_Type = Classification of Industry affiliation (1=Industry A, 0=other) Buy_Sit = Type of buying situation (NEW = New, MOD = Modified purchase, REP = Repeat Purchase

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL Using the COMP1 dataset, determine the following descriptive statistics for each quantitative variable: The most appropriate measurement of central tendency; Two measurements of dispersion; For one variable, identify if any outliers exist. Determine these statistics using the f(x) options: =AVERAGE, =MEDIAN, =MODE, =STDEV, =VAR

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Using the COMP1 dataset, determine the following descriptive statistics for each quantitative variable: The most appropriate measurement of central tendency; Two measurements of dispersion; For one variable, identify if any outliers exist. This time, use the “Descriptive Statistics” option: TOOL DATA ANALYSIS DESCRIPTIVE STATISTICS MODULE 1: Data Analysis Using EXCEL

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL Using the COMP1 dataset, develop a histogram for the quant variable of your choice (use three categories). EXAMPLE: Take the Usage variable and subtract the min value from the max value (65-25 = 40). Divide that number (the range) by the number of desired categories (40/3 = 13.33). Now, “massage” that figure as necessary to create reasonable, logical categories of approximately equal size: 25-39 40-54 55-65

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL A few points about histograms in EXCEL: Create a column over to the right that contains only the TOP of each category that you have assigned Label this category “BIN RANGE” TOOLS DATA ANALYSIS HISTOGRAM Ensure that the “Labels” box is ticked Ensure that the “Cumulative Percentage” and “Chart Output” boxes are ticked

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL Using the COMP1 dataset, develop a scatterplot of two quantitative variables of your choice. EXAMPLE: Chart the two variables Usage and Price Level. NOTE: it is helpful to move the two variables into columns next to each other

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL Using the COMP1 dataset, develop a pivot table of the entire dataset. DATA PIVOT TABLES NOTE: ensure that the labels are included in the table

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL A few more widely used examples of functions in EXCEL: =IF =AND =ABS =SUMPRODUCT =RAND =RANDBETWEEN

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 1: Data Analysis Using EXCEL Look up tables in EXCEL can be a very helpful way of converting quantitative data into categorized qualitative data…which sometimes can be easier to work with. Use the VLOOKUP function to categorize the Speed variable into “EXCELLENT” “AVERAGE” and “POOR” events. Clarification of EXCEL notation : VLOOKUP(enter the speed value here,enter the range of the two column table here,enter the column number of the desired category label here)

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Lagniappe : This word derives from New World Spanish la ñapa, “the gift,”. The word came into the Creole dialect of New Orleans and there acquired a French spelling. It is still used in the Gulf States, especially southern Louisiana, to denote a little bonus that a friendly shopkeeper might add to a purchase. One Lagniappe presented here is the process of selecting “the best” option from among several alternatives.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Lagniappe for everyone!

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Statistics Workshop Topics MODULE ONE: Concept Review Review of Statistical Concepts Data Analysis using EXCEL MODULE TWO: Inferential testing using EXCEL and SAS Confidence Intervals Ttests ANOVA Chi-Square MODULE THREE: Predictive Modeling using EXCEL and SAS Regression Analysis Logistic Analysis Discriminant Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals We always prefer to use descriptive statistics. However, often we are forced to take a sample and use inferential statistics because of issues related to cost, time, money or access. When taking a sample, we can estimate a population parameter such as a mean or a proportion using the sample statistic (which is not very accurate) or we can calculate a confidence interval.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Confidence Intervals are calculated using two formulas: CI for the population mean: x + Z*(s)/SQRT(n) where, x = the sample mean Z = Z-score (90%CI = 1.645, 95%CI = 1.96, 99%CI = 2.575) s = sample standard deviation n = sample size Note: the part of the expression after the + is called the Margin of Error Z*(s)/SQRT(n). MODULE 2: Inferential Testing – Confidence Intervals

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The CI for the population proportion is represented by: p + Z*SQRT((pq)/n) where, p = the sample proportion Z = Z-score (90%CI = 1.645, 95%CI = 1.96, 99%CI = 2.575) q=1-p n = sample size Note: the part of the expression after the + is called the Margin of Error Z*(s)/SQRT(n). MODULE 2: Inferential Testing – Confidence Intervals

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals Fun Manual Calculation! using the Gallup Website: http://poll.gallup.com/ Replicate the Gallup prediction using the second CI formula.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals More fun calculations! Using the Comp1 dataset, calculate the 95% CI for a quantitative variable using formula 1. =CONFIDENCE Alpha is the accepted prob. of making a T1 error. It is also 1-the confidence level.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals The result generated from =CONFIDENCE(.05, 1.32, 100) is 0.258857. What does this number mean? This is the Margin of Error. In other words, if you were to report the 95% confidence interval for this company’s speed, you would report: 3.52 +.25 Or 3.25 to 3.77

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Lets execute CIs in SAS. First, calculate the 95% CI for the quantitative variable Speed. Use the following code: Proc Import datafile = "c:\COMP1.xls" OUT = COMP1 DBMS = "EXCEL97" Replace; run ; Proc Print data=Comp1; Run ; Proc Means data=Comp1 CLM alpha=.05 ; Var Speed; Run ; MODULE 2: Inferential Testing – Confidence Intervals Should you require a higher or lower alpha (.01 is more conservative and.10 is more risk tolerant), change the.05 as appropriate.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals The output is pretty simple: Notice that this is the same interval from the EXCEL output. Isnt it nice when numbers match?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Now, calculate the 95% CI for the qualitative variable Procure. Use the following code: Proc Freq data=Comp1; Tables Procure/Binomial alpha=.05 ; Run ; MODULE 2: Inferential Testing – Confidence Intervals Note that EXCEL does not readily support this calculation.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing – Confidence Intervals The output is pretty simple:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. Quantitative Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The Chi-Square Test is used to determine if two QUALITATIVE variables are related. The Chi-Square statistic is computed using the following formula: X 2 = Σ(f o -f e ) 2 /f e Where: f o is the frequency of the observed value f e is the frequency of the expected value This calculated test statistic is converted into a p-value to evaluate the presence of a relationship (or not). MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Using the COMP1 dataset, determine if a company’s buying situation (BuySit) is related to the centralization of their purchasing decisions (Procure). The hypothesis statements for this test are: H o : BuySit and Procure are NOT related H 1 : BuySit and Procure ARE related Develop the appropriate testing matrix and identify the Type1 and Type2 errors. MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Now, using EXCEL, develop the 2x3 matrix of these two variables using a pivot table (place “count of procure” in the center). You should see this: This table includes the “frequencies of the observed” values from our Chi-Square formula. MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Now, we need to determine the expected values. If there is NO relationship, then we would expect to see exactly 32% of the decentralized procurement companies with a modified buying situation, 34% with a new buying situation, etc. The matrix of expected values looks like this: MODULE 2: Inferential Testing =(32/100)*50

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Now use the =CHITEST formula in EXCEL. The actual range requested is the INTERIOR of the observed matrix and the expected range is the INTERIOR of the expected matrix (note that the marginal values are the same for the two matrices). MODULE 2: Inferential Testing The resulting value is: 1.61E-09 or.00000000161.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics This is a p-value. Given the rule : if p a accept Ho… Can we conclude that there is a relationship between the centralization of the purchasing decision and the buying situation? MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Lets execute the same test in SAS, using the following code: MODULE 2: Inferential Testing Proc Freq data=Comp1; Tables Procure*BuySit/CHISQ; Run ;

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics This code creates the following output: MODULE 2: Inferential Testing This legend provides the key to interpreting these numbers

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Here is the second part of the output: MODULE 2: Inferential Testing This is the same p-value that we obtained in EXCEL

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Any additional questions on Chi-Square?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. Quantitative Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Ttests represent the most common form of statistical testing. It involves either one sample or two independent samples. 1.One Sample Ttest - compares the mean of the sample to a given number. e.g. Is average monthly revenue per customer who switches >$50 ? Formal Hypothesis Statement examples: H 0 : $50 H 1 : > $50 H 0 : = $50 H 1 : $50 MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics 2.Two Sample Ttest - compares the mean of the first sample minus the mean of the second sample to a given number. e.g. Is there a difference in the production output of the two facilities? Formal Hypothesis Statement examples: H 0 : a b H 1 : a > b H 0 : a = b H 1 : a b MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics 3.Paired Sample Ttest - compares the mean of the differences in the observations to a given number. e.g. Is there a difference in the production output of a facility after the implementation of new procedures? Formal Hypothesis Statement example: H 0 : post - pre <=0 H 1 : post - pre > 0 MODULE 2: Inferential Testing

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using EXCEL Using the COMP1 dataset and EXCEL, determine if there is a difference in the overall satisfaction of large versus small companies (the large companies will represent a sample and the small companies will represent a second sample). Determine the null and alternative hypothesis statements for this question…then develop the 2x2 hypothesis matrix…including the Type1 and Type2 errors. Now, sort the data by size. The satisfaction values associated with the 0s (the small firms) will be our first array of numbers and the satisfaction values associated with the 1s (the large firms) will be our second array of numbers.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using EXCEL Satisfaction of small firms Satisfaction of large firms Two tailed test homoscedastic Your computed value should be 1.80363E-06…what do you conclude?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using SAS Using the COMP1 dataset and SAS, determine if there is a difference in the overall satisfaction of large versus small companies. Here is the necessary code: Proc Ttest data=Comp1; Var Satisf; Class Size; Run ; the quantitative variable of interest the qualitative variable which identifies the two samples

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using SAS

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using SAS Using the COMP1 dataset and SAS, determine if the overall satisfaction is more than 5.0. Develop the hypothesis statements and develop the 2x2 matrix. Here is the necessary code: Proc Ttest data=Comp1 H0=5.0; Var Satisf; Run;

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing using SAS

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Any additional questions on Ttests?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. Quantitative Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA As with Ttests, ANOVA is a common test to determine if differences exist among samples. Where Ttests evaluate either one or two samples (groups), ANOVA accommodates 3 or more. The hypothesis statements in ANOVA look like this: Note: the hypotheses are interpreted as “at least one mean is different”…not that all means are different. H 0 : a = b = c H 1 : a = b = c

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA SST = SSW + SSB i j (X ij -X) 2 = i j (X ij -X j ) 2 + n j (X j -X) 2 SST = Total Sum of Squares SSW = Sum of Squares Within Groups SSB = Sum of Squares Between Groups X = mean of data for all the sample groups combined X j = mean of the jth sample group X ij = the ith element from the jth group n = number of samples in each group _ _

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics FUN MANUAL CALCULATION OF ANOVA!

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Question: If we have to test for a difference among a group of three or more samples, why cant we simply execute a series of ttests? Hint: the issue has to do with the accepted risk of making a TYPE 1 error.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA Using the Comp1 dataset in EXCEL, determine if there is a difference in satisfaction among the different types of buying situations. Hint: we will first have to convert the values for the variable “buying situations” from alpha characters to numeric characters…the data is still categorical…EXCEL cannot read alpha characters for ANOVA. =IF(M2="NEW",1,IF(M2="MOD",2,IF(M2="REP",3))) Then: TOOLS DATA ANALYSIS ANOVA: SingleFactor Note: the input range will be the quant variable and the categorical (numeric) variable ONLY. And…it helps if they are next to each other.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA Your output should look like this: This is your p- value. What do you conclude? SSB SSW

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Question: Where are the differences?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA Using the Comp1 dataset in SAS, determine if there is a difference in satisfaction among the different types of buying situations. Here is the necessary code: Proc ANOVA data=Comp1; Class Buysit; Model Satisf=Buysit; Means Buysit/TUKEY; Run; Qualitative Independent/Classification Variable Quantitative Dependent=Independent Statement to determine WHICH groups are different

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA Note that this output looks almost the same as the EXCEL output SSB SSW

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing - ANOVA

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 2: Inferential Testing Any additional questions on ANOVA?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Statistics Workshop Topics MODULE ONE: Concept Review Review of Statistical Concepts Data Analysis using EXCEL MODULE TWO: Inferential testing using EXCEL and SAS Confidence Intervals Ttests ANOVA Chi-Square MODULE THREE: Predictive Modeling using EXCEL and SAS Regression Analysis Logistic Analysis Discriminant Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. QuantitativeQuantitative or Dummy Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable Categorical >2 Categories Quantitative or Dummy Discriminant Analysis Test establishes a discriminant model; used to explain, predict or control dependent variable BinaryQuantitative or Dummy Logistic AnalysisTest establishes a logistic model; used to explain, predict or control dependent variable MODULE 3: Predictive Modeling

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling All models are wrong…but some are useful.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Correlation Analysis Prior to any model development, an initial correlation analysis is typically generated to understand which variables are related. As with any inferential method, Correlation Analysis is conducted within the context of Hypothesis Statements. The general form of these statements in Correlation Analysis is: H0: Variable A and Variable B are NOT related H1: Variable A and Variable B ARE related so, if p<a, then the conclusion is that Variable A and Variable B ARE related.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Determine which variables in the Comp1 dataset are correlated using EXCEL. TOOLS DATA ANALYSIS CORRELATION MODULE 3: Predictive Modeling – Correlation Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The output is below. What do the numbers mean? MODULE 3: Predictive Modeling – Correlation Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics These numbers represent the percentage of change in one variable that moves with the change in another variable. Because these values are percentages, they can vary from a low of negative 1 to a high of positive 1, including 0. For example, Price Level and Speed have a correlation of -.34923. This means that 35% of the change in Price Level moves with 35% of the change in Speed…and vice versa. In addition, since this number is negative, the change moves in the OPPOSITE direction (i.e. as one goes up, the other goes down). Note: Correlation does NOT equate to causation. At this point in the analysis, we cannot state that Price Level decreases Speed or vice versa. We can only claim that we know that they have an inverse relationship. MODULE 3: Predictive Modeling – Correlation Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics The scatterplot below is a visual representation of the correlation coefficient -.34923. Is this relationship significant? MODULE 3: Predictive Modeling – Correlation Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics SAS provides more information regarding the statistical significance of any correlations. The statistical significance will be highly sensitive to the sample size…if the sample is too large…EVERYTHING will appear to be significant. Here is the necessary code: Proc Corr data=Comp1; Run; MODULE 3: Predictive Modeling – Correlation Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Correlation Analysis Note: the number on the top is the correlation coefficient and the number on the bottom is the p-value

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Correlation Analysis Rather than execute a full matrix, if you are only interested in assessing the correlations of the variables with one or two variables of interest (e.g., dependent variables), include the “with” option: Proc Corr data=Comp1; With Usage; Run;

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Correlation Analysis So…what will you conclude?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression In Regression, we generate a line of the form: Y = β 0 + β 1 X 1 + β 2 X 2 +… β n X n that represents “best-fit” of the data…meaning that the difference between the actual and the predicted values is minimized. In this equation, Y = the predicted value (MUST be a ratio scale variable) B 0 = the Y intercept B 1 = the coefficient or weight of X1 X 1 = an independent or predictor variable (MUST be a ratio scale variable or a dummy variable)

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics To evaluate the “ Best fit line ”, we calculate the following statistics using ANOVA (again usually done by the computer program): Total sums of squares (SST), Σ (Y i - Y) 2 is a measure of the variability of the dependent variable Y Error sums of squares (SSE), Σ (Y i - Y i -Pred) 2 is a measure of the variability of the dependent variable Y that is left over after using the regression model and the predictor variable X to explain Y. If this value becomes 0, then our Regression model is perfect (usually not the case!) MODEL(Regression) sums of squares (SSR), SST - SSE, is a measure of the amount of the total sums of squares accounted for by the regression model. The closer SSR is to SST, the better the model explains the variation of Y. Coefficient of determination, or R 2, is calculated as SSR/SST. This is the proportion of the total sums of squares (i.e. total variation in Y) that is explained by the regression model. R 2 is close to 1 for good fits and close to 0 for no relationship. MODULE 3: Predictive Modeling – Regression

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression Lets execute a Regression analysis in EXCEL. Use Usage as the dependent (y) variable. TOOLS DATA ANALYSIS REGRESSION

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression The dependent variable is the Y Range All other quantitative or dummy variables are in the X Range The Residuals can be helpful

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression From this output, what do you know?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression Using this output, answer the following questions: 1.What is the linear equation? 2.Which predictor variables are significant? 3.How accurate is the model? 4.How could you improve the efficiency (parsimony) of the model? 5.Can you use this model to make a prediction? 6.Are there any problems with the model? Such as multicollinearity among the predictors? Or any influential observations?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression Although a Regression model can be developed using EXCEL, the process is fairly manual and there are limited diagnostics.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression Limitations of using EXCEL for Regression modeling: 1.Only supports an “all in” selection method; retention or deletion of variables must be done manually. 2.No diagnostics for collinearity among the predictors. 3.No intervals for the predictions. 4.Cumbersome identification of outliers.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression Proc Reg data=Comp1; Model Usage = SpeedPriceLvPriceFlx Man_Imag Service Sal_Imag Quality Size Satisf SpecBuy Procure IndType/ VIF P R CLI Selection=Forward Partial; Output out=reg p=pred; Plot Residual.*Pred.; Run ; Quit; Here is the necessary code:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Regression

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Looks as if Obs 7 might be a problem for us… MODULE 3: Predictive Modeling – Regression

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Any additional questions on Regression?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. QuantitativeQuantitative or Dummy Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable Categorical >2 Categories Quantitative or Dummy Discriminant Analysis Test establishes a discriminant model; used to explain, predict or control dependent variable BinaryQuantitative or Dummy Logistic AnalysisTest establishes a logistic model; used to explain, predict or control dependent variable MODULE 3: Predictive Modeling

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Discriminant Analysis What if we need to create a model with a CATEGORICAL dependent variable? Regression Analysis is not a viable option. ANOVA should be executed first. Followed by Discriminant Analysis.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Proc Discrim data=Comp1 method=normal Pool=Yes List Crossvalidate; Class BuySit; Var Speed PriceLv PriceFlx Man_Imag Service Sal_Imag Quality Size Satisf SpecBuy Procure IndType Usage; Run ; MODULE 3: Predictive Modeling – Discriminant Analysis Here is the necessary code to execute Discriminant Analysis:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Discriminant Analysis The objective in Discriminant Analysis is to minimize the errors in the “Hit Matrix”:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Dependent Variable Independent (predictor) Variable Hypothesis TestComments Categorical (Qualitative) Chi-SquareTests if variables are statistically independent (i.e. are they related or not?) QuantitativeCategorical (Qualitative) T-TEST ANOVA Determines if categorical variable (factor) affects dependent variable; Ttests for 1 or 2 groups and ANOVA for 3 or more. QuantitativeQuantitative or Dummy Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable Categorical >2 Categories Quantitative or Dummy Discriminant Analysis Test establishes a discriminant model; used to explain, predict or control dependent variable BinaryQuantitative or Dummy Logistic AnalysisTest establishes a logistic model; used to explain, predict or control dependent variable MODULE 3: Predictive Modeling

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Logistic Analysis Logistic Analysis is a particularly handy modeling technique when the dependent variable can assume two (and only two) values. Common Applications of Logistic Analysis: Will an individual qualify for a loan? Does this person have a particular disease? Will this person respond to a marketing solicitation? Will applicant A be accepted to college? Will the product fail in a particular situation?

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Logistic Analysis The general form of the Logistic Equation is: Prob (event) = e B0 + B1X1+B2X2…+BnXn Prob (no event) The objective in Logistic Analysis is to correctly predict the presence of an event. The accuracy of the model, is then evaluated using several metrics, including a classification table.

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics Proc Logistic descending data=Comp1; model SpecBuy = Speed PriceLv PriceFlx Man_Imag Service Sal_Imag Quality/ Selection=Stepwise CTable PPROB = ( 0 to 1 by.1 ) Lackfit RISKLIMITS; run ; quit; MODULE 3: Predictive Modeling – Logistic Analysis Here is the code to execute Logistic Analysis:

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Logistic Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics MODULE 3: Predictive Modeling – Logistic Analysis

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c 2006 Jennifer Priestley, Ph.D. Kennesaw State University Department of Mathematics and Statistics General Questions on Predictive Modeling?

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