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One-WayAnalysis of Variance Chapter 12,Section 1 Statistical Methods II QM 3620
Suppose that we would like to compare two or more means from different populations. Example: A business would like to determine if the average purchase is different during different months of the year to see if staffing should be adjusted. Example: A university wants to know if the average grade given by one professor is higher than those of other professors. More than one group needs to be compared. We only have a sample of information from each group. You are being asked to make judgments about different groups based on only limited information from each group. …And remember,we are making judgments about the entire group (i.e. the population), not just the sample we collected. Back to Quantitative Variables
One-Way ANOVA –The Purpose To compare the means of two or more populations,we can use a quick and relatively painless (yet very powerful) approach called ANOVA (or ANalysis Of VAriance). We are still going to rely on the normal distribution, mainly because samples are usually not that small and it provides a very slick and pretty robust approach. Remember, the Central LimitTheorem kicks in when the sample sizes are 30 or more,and actually has quite a profound effect well before the sample size even reaches 30. Subsequent modules will address the comparison of two or more groups on qualitative variables (like proportions in a category)
One-Way ANOVA – How it Works Let’s consider two situations concerning the average amount spent on groceries in a month by families: In situation 1,we have separated families into three groups based on their socio-economic status. The three groups in this situation are lower-class families, middle-class families, and affluent families. In situation 2,we have separated families into three groups based on the month in which the father was born: Jan-Apr, May-Aug, and Sep-Dec. Okay, it sounds really silly,but it will help make the point.
One-Way ANOVA – How it Works Do the groupings matter … in terms of the variable measured (that would be the average amount spent on groceries in a month)? It is very likely that the groups based on economic-status will have different averages for the amount they spend. In essence,these are three different populations with very different overall average spending levels. The birth month of the father in the family probably has no little or no bearing on the average amount spent on groceries. In essence, these groups would all have about the same average spending as they are really just arbitrary groups from one population.
One-Way ANOVA – How it Works So,if we took a sample from each group in each situation … The averages for the three economic groups will not appear as if they could have come from the same population. Remember, even if samples are taken from one population, the means will change from sample to sample. If the samples are from truly different populations, then the average would be more different than what we would expect if they were just random samples from the same population. In the latter situation, the groups really are not different. Therefore, even though the samples will have different means, they will appear as simply different samples from the same population.
One-Way ANOVA –The Hypothesis All population means are equal. We use μ k at the end to represent the mean of the last group. You are comparing k groups with this test. The null hypothesis is our“fallback” position and is assumed to be true until proven otherwise. The alternative hypothesis states that the means in each of the groups are not all the same. The mean for at least one of the groups is different than the mean of some other group IN THE POPULATION. We use μ i to represent the mean of any of the groups. The letter i could be any value from 1 to k,where k is the number of groups. Saying that all of the population means are not the same is not saying that all population means are different (some may be the same.) We will need to find which ones are different in a later step. μ k μ2μ2 H0 : μ1H0 : μ1 H A :Not all μ i are the same
μ3μ3 μ2μ2 μ1μ1 μ3μ3 μ2μ2 μ1μ1 One-Way ANOVA – Situation 1 All POPULATION Means are not the same: The Alternative Hypothesis isTrue or or something like that
One-Way ANOVA – Situation 2 All POPULATION Means are the same: The Null Hypothesis isTrue μ3μ3 μ2μ2 μ1μ1
How it Works Graphically (simplified) The distributions are of the amount of money spent by a family for groceries. x1x1 x2x2 x3x3 x 1 x 2 x 3 Situation 1 Samples from different groups Situation 2 Different samples from the same group
x1x1 x2x2 x3x3 Situation 1 Variation within the groups small compared to variation between the sample means Situation 2 Variation within the group large compared to variation between the sample means x 1 x 2 x 3 How it Works Graphically (simplified) So what is the difference?
x1x1 x2x2 x3x3 Situation 1 Variation within the groups small compared to variation between the sample means Situation 2 Variation within the group large compared to variation between the sample means x 1 x 2 x 3 How it Works Graphically (simplified) So what is the difference? Since we are comparing variation within the groups to the variation between the groups (implied from the sample means), we actually find differences in the means of populations by analyzing variation. Hence the Analysis of Variance (ANOVA)
Remember the Process 1) 2) 3) 4) 5) 6) Specify population parameters of interest Formulate the null and alternative hypotheses Specify the desired significance level, Take a random sample from each group and calculate relevant statistics Calculate p-value for test and compare it to Reach a decision and draw a conclusion
Remember, our whole reason for this series of thoughts and calculations was to help us determine if the means of different groups are the same or different by using only the limited information provided by a sample from each group. SoWhat was the Point of All This?
Let’s try this for real ApplicationTime
Bayhill Marketing Company designs coupons for their clients. Their current client will be sending these coupons to American Express customers in their monthly statement. Bayhill formed focus groups and u sed their input to design four coupons. To evaluate which of these coupons is the most effective, Bayhill sent each of these coupons to randomly selected American Express cardholders. Data from the redemption of these coupons at the client were collected. It was decided to define“effective” coupons as those that provided the highest dollar values of associated orders. The coupon design that provided the highest average order would be selected. How can the company determine the most effective coupon design to use? Business Application Highlights
We are going to compare the average dollar redemption for each coupon design and determine if one design has a distinguishably higher average. We infer about the greater population of customers from our limited sample of information. We want to be fairly certain of our decision despite our lack of information. Thus,we will want compelling evidence to support our choice of coupon design. The Approach
One-way analysis of variance is a statistical approach used to find differences between two or more groups by analyzing the variance among the observations within a group versus the variance between the groups. The factor is how the groups that are to be compared are defined. In this case, the coupon design defines the different groups. Each group has a different coupon design. The levels can be thought of as the specific group “names” or how the groups are defined on the factor. In this case, the levels are the specific coupon designs (referred to generically in the problem as Design #1, Design #2, etc.) Terminology
A completely randomized design is when the levels are randomly assigned to observations in the study. In this case, the coupon designs were sent randomly to individuals. Individuals were not preselected to receive a certain coupon because of some characteristic. A balanced design is when the levels all have the same number of observations. In this case, a sample of eight was taken from each coupon design redemption. This study could just have easily been an unbalanced design. There is no real reason that we had to take the same amount of observations from each group. Terminology
The within-sample variation refers to the variation among the observations within a sample. In this case, the dollar value of the coupon redemption was not the same for each observation using a certain coupon design. This variance within a coupon design is the within-sample variation. The between-sample variation is how much the averages for each group vary. Thus, although the individual redemptions vary within a coupon design (i.e. the within- sample variation),the average redemption for each coupon design varies among the four coupon designs (i.e. the between-sample variation). Terminology
The Null and Alternative Hypothesis H 0 : All means are the same (equal) H A : At least two means are different The Alpha Level:.05. The p-value of the test will tell us the“likelihood” that the null hypothesis is true. Low p-values (at or below the alpha level) indicate evidence to support some difference among the coupon design averages (i.e.the alternative hypothesis). High p-values (above the alpha level) indicate that there was not enough evidence to show a difference. Using Statistics
. How do the mathematics decide between the hypotheses? If the variance among the group means is more than can be explained by the variance from the individual observations alone, then there must be a difference among the group means. Hence, this analysis, while determining difference among means, is really based on differences among variances. That is why it is called ANOVA or ANalysis Of Variance. Pages go into more detail on how this analysis takes place technically. Using Statistics
Populations are normally distributed. Although ANOVA assumes normal distributions,it is quite robust (able to handle deviations) to this assumption. As long as the observations do not show extreme non-mound-shaped distributions, this assumption will probably be okay. Population variance are equal. Here is where the balanced design comes in handy. As long as we have about the same number of observations in each group as in our current situation, ANOVA is quite robust to this assumption. If our samples sizes are very different, we will have ensure that unequal variances are not a problem. The textbook demonstrates the Hartley’s F-Test for equal variance on pages Note that this test requires the first assumption – normality of observations – to be truly applicable. The Assumptions - Part 1
Sampled observations are independent. In this case,we are assuming that the redemption of each coupon was not affected by the redemption of another coupon. There is really no easy way to check this assumption. Just think about how things are being transacted… Data are interval or ratio level. Basically, this assumption is about whether or not an average actually means anything. When using ratio or interval data, the average makes sense. You can not take an average of categories, and averages of rankings are suspect (but are often done anyway.) TheAssumptions - Part 2
Select the largest of the variations… we’ll call it s max just to make things Select the smallest of the variations … we’ll call it s min so we are consistent s max s min Hartley’s F-Test allows us to do a check of the equal variance assumption required byANOVA. Basically, the test compares the largest sample variation to the smallest sample variation among the groups. To do the test,we first need the variation of the observations in each of the group samples. The good news is that Excel kicks this out as a standard part of the output of an ANOVA report. difficult. with our prior difficulties. Calculate the ratio of the two and refer to it as If F max is smaller than the value from the table that Dr. Hartley so generously provided, in Appendix I in the text, then there is no reason to be concerned with the assumption of equal variance. Hartley’s F-Test of Equal Variance 2222 F max 2 2
Hartley’s F-Test of EqualVariance First, calculate Fmax To find the value in Dr. Hartley’s table, you will need three values: α - which is the same alpha level you are using on the ANOVA k - which is the number of groups I – average sample size in the groups minus one Look up the number in the Excel table that matches the row and column titles. If you really insist on using the table in the textbook, then set c to k, and v to ( average sample size - 1). They relabeled the columns and rows to confuse you. Compare Fmax to the value from the table. If it is smaller than the table value, you may proceed.
Using the data file link below,save the file,and then open the file in Excel Bayhill Bayhill – Data file file Appendix Appendix I and J in Excel I and J in Excel Video Demonstration of theAnalyses ANOVA Hartley’s Hartley’s F-Test When you are finished with the above steps,you should have a completed analysis. Be sure to add explanatory comments and documentation to your file. For comparison,access the completed file below: Bayhill Bayhill – Solution file Solution file Remember:Press and hold the CTRL button when you click on a file link Doing theAnalysis
The Hydronics Corporation (pg. 523) makes and distributes weight loss supplements. Hydronics must test to see if the products are effective prior to distribution. Two herb-based products are under consideration. To gauge effectiveness, Hydronics includes a placebo in this study. 300 human subjects have volunteered for the study, with 1/3 being randomly assigned to each of the products (Product 1 and Product 2) and the placebo. The study takes place over six weeks. Subjects are unaware of what product they are using (i.e. blind study). Business Application Highlights
Some of the subjects did not make it through the entire six weeks, so the eventual number of observations differed from the original 100 subjects per product. Hydronics wants to determine if the two products are more effective than doing nothing (i.e. the placebo) and whether there is a difference even among the products. The weight loss after six weeks was recorded for each individual that remained in the study. Positive numbers in the data set indicate weight loss. Negative numbers indicate a weight gain. Business Application Highlights
We are going to compare the averages weight loss for each product (including the placebo) and determine if any of the products has a significantly higher average loss. We can only gather volunteers for this study and have to rely on them to accurately report their use of the product. We also hope that they do not change their normal behavior as far as it affects their food consumption. Although we only have volunteers,we are going to use the results to make inferences about the greater population,or at least the population that would be in the market for a weight loss product. The Approach
The factor in this case is the product (you might also call it the pill or application.) Basically, individuals received either a placebo (sugar pill) or one of two products in pill form. The levels are the specific products: the placebo, Product 1 or Product 2. There are three levels or groups. This application is an unbalanced design as there are slightly different sample sizes in each of the groups. The within-sample variation would refer to the variation in weight loss among members in a group. A ll individuals that took the placebo did not lose exactly the same weight. The variation in their losses (and those within the Product 1 and Product 2 groups) would be the within-sample variation. The between-sample variation in this case is the variation in the mean weight loss between theplacebo group, the Product 1 group, and the Product 2 group. Applying theTerminology
The Null and Alternative Hypothesis Ho: there is no difference in the average weight loss for the three products Ha: there is a difference in the average weight loss for the three products If some difference is found between the products,the obvious next step would be to determine where a difference exists. ANOVA simply tells you there is some difference and the alternative hypothesis does not state that every mean is different,just that at least two means differ. To find the specific differences, we will utilize a technique called theTukey-Kramer Procedure for Multiple Comparisons. Using Statistics
Populations are normally distributed. Population variance are equal. Sampled observations are independent. Data are interval or ratio level. Evaluating the Assumptions
The Concept A group can only be compared to one other group at a time. There is really no feasible way to compare three or more groups simultaneously. For that reason,we make pairwise comparisons (two groups at a time.) The Experiment-Wide Error Rate To control for this,we use a technique that allows us to make multiple pairwise comparisons and still keep the overall chance of an alpha error at the stated level. The Calculations There are some tricks in Excel that will make this calculation easier, but you will need to focus carefully. TheTukey-Kramer procedure will ultimately provide you with critical ranges that must be compared to each pairwise difference. Differences in excess of these critical ranges are deemed significant. Tukey-Kramer Comparison Procedure
Tukey-Kramer Calculation 1nj1nj MSW 1 2 n i qCriticalRange where: q = Value from Appendix J in the textbook using k (across the columns) and n T - k (down the rows), where k = number of groups, and n T = total number of observations Use page 954 for= 0.05 and page 955 for= 0.01 MSW = MS Within Groups from ANOVA table in Excel analysis n i and n j = Sample sizes from groups i and j
Tukey-Kramer Example Group Group Group x3x3 x2x x 1 x 2 x 1 x 3 1.Compute absolute differences between sample means: x x x 3 2.Find the q value inAppendix J. Use page 954 as= Since k = 3 groups and n T = 15 total observations,n T - k = 12. Look up the value in the column labeled“3” and row labeled “12” qαqα
5. All of the differences in sample means are greater than the Critical Range. Therefore we can say that there is a difference between the population means of each group CriticalRange 1nj1nj MSW 1 2 n i qαqα Tukey-Kramer Comparison 3. Compute Critical Range (MSW would come from the ANOVA table): x3x3 x2x2 x 1 x 2 x 1 x 3 4. Compare the differences from the previous slide to the Critical Range:
Using the data file link below,save the file,and then open the file in Excel Hydronics – Data file Hydronics Data file Video Demonstration of theAnalyses ANOVA ANOVA Hartley’s F-Test Hartley’s Looking up theTukey-KramerTableValue Looking up theTableValue Setting up theTukey-Kramer ComparisonTable Setting up the ComparisonTable DeterminingTukey-Kramer Significant Differences Determining Significant Differences When you are finished with the above steps,you should have a completed analysis. Be sure to add explanatory comments and documentation to your file. For comparison,access the completed file below: Hydronics – Solution file Hydronics Solution file Remember: Press and hold the CTRL button when you click on a file link Doing theAnalysis
ADDITIONAL PRACTICE Westfall Example,Page 521 Digitron Example,Page 527 Problem 12-14,Page 534 Data Solution