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QNT 531 Advanced Problems in Statistics and Research Methods

QNT 531 Advanced Problems in Statistics and Research Methods. WORKSHOP 2 By Dr. Serhat Eren University OF PHOENIX. SECTION 2. ANALYSIS OF VARIANCE AND EXPERIMENTAL DESIGN. SECTION 2 SECTION OBJECTIVES. An Introduction to Analysis of Variance

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QNT 531 Advanced Problems in Statistics and Research Methods

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  1. QNT 531Advanced Problems in Statistics and Research Methods WORKSHOP 2 By Dr. Serhat Eren University OF PHOENIX

  2. SECTION 2 ANALYSIS OF VARIANCE AND EXPERIMENTAL DESIGN

  3. SECTION 2SECTION OBJECTIVES • An Introduction to Analysis of Variance • Analysis of Variance: Testing for the Equality of k population means • Multiple comparison procedures • An introduction to Experimental Design • Completely Randomized Designs • Randomized Block Design • Factorial Experiment

  4. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS One-Way Designs: The Basics • A factoris a variable that can be used to differentiate one group or population from another. It is a variable that may be related to he variable of interest. • A levelis one of several possible values or settings that the factor can assume. • The response variableis a quantitative variable that you are measuring or observing.

  5. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • These are all examples of one-way or completely randomized designs. • An experiment has a one-way or completely randomized designif there are several different levels of one factor being studied and the objects or people being observed/ measured are randomly assigned to one of the levels of the factor.

  6. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • The term one-wayrefers to the fact that the groups differ with regard to the one factor being studied. • The term completely randomizedrefers to the fact that individual observations are assigned to the groups in a random manner.

  7. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS Understanding the Total Variation • Analysis of variance (ANOYA) is the technique used to analyze the variation in the data to determine if more than two population means are equal. • A treatmentis a particular setting or combination of settings of the factor(s)

  8. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • The grand mean or the overall mean is the sample average of all the observations in the experiment. It is labeled (x-bar-bar). • Now we can rewrite the variance calculations as follows:

  9. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • The total variation or sum of squares total(SST) is a measure of the variability in the entire data set considered as a whole. • SST is calculated as follows:

  10. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS Components of Total Variation • The between groups variation is also called the Sum or Squares between or the Sum of Squares Among and it measures how much of the total variation comes from actual differences in the treatments. • The dot-plot shown in Figure 14.3 displays the sample average for each of the four time treatments. These are called treatment means.

  11. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • A treatment meanis the average of the response variable for a particular treatment. • Between Groups Variationmeasures how different the individual treatment means are from the overall grand mean. It is often called the sum of squares between or the sum of squares among (SSA).

  12. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • The formula for sum of squares among (SSA) is: • Within groups variation measures the variability in the measurements within the groups. It is often called sum of squares within or sum of squares error (SSE).

  13. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS The Mean Square Terms in the ANOVA Table • The mean square amongis labeled MSA The mean square erroris labeled MSE and the mean square totalis labeled MST. • The formulas for the mean squares are;

  14. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS Testing the Hypothesis of Equal Means • In general, the null and alternative hypotheses for a one-way designed experiment are shown below: HA: At least one of the population means is different from the others.

  15. SECTION 2ANALYSIS OF DATA FROM ONE-WAY DESIGNS • The formula for the F test statistic is calculated by taking the ratio of the two sample variances: • In ANOVA, MSA and MSE are our two sample variances. So the F statistic is calculated as:

  16. SECTION 2ASSUMPTIONS OF ANOVA • The three major assumptions of ANOVA are as follows: • The errors are random and independent of each other. • Each population has a normal distribution. • All of the populations have the same variance.

  17. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS • Ablockis a group or objects or people that have been matched. Are object or person can be matched with itself, meaning that repeated observations are taken on that object or person and these observations form a block? • If the realities of data collection lead you to use blocks, then you must take this into account in your analysis. Your experimental design is called a randomized block design. Instead of using a one-way ANOVA you must use a block ANOVA.

  18. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS • An experiment has a randomized block designif several different levels of one factor are being studied and the objects or people being observed/ measured have been matched. • Each object or person is randomly assignedto one of the c levels of the factor.

  19. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS Partitioning the Total Variation • Like the approach we took with data from a one-way design, the idea is to take the total variability as measured by SST and break it down into its components. • With a block design there is one additional component: the variability between the blocks. It is called the sum of squares blocks and is labeled SSBL.

  20. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS • The sum of squares blocks measures the variabilitybetween the blocks. It is labeled SSBL. • For a block design, the variation we see in the data is due to one of three things: the level of the factor, the block, or the error.

  21. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS • Thus, the total variation is divided into three components: SST = SSA + SSBL + SSE

  22. SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS Using the ANOVA Table in a Block Design • The ANOVA table for such a block design looks just like the ANOVA table for a one-way design with an additional row.

  23. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS Motivation for a Factorial Design Model • An experimental design is called a factorial design with two factorsif there are several different levels of two factors being studied. • The first factor is called factor Aand there are rlevels of factor A. The second factor is called factor Band there are c levels of factor B.

  24. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • The design is said to have equal replicationif the same number of objects or people being observed/measured are randomly selected from each population. • The population is described by a specific level for each of the two factors. Each observation is called a replicate. • There are n'observations or replicates observed from each population. There are n = n'rcobservations in total.

  25. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS Partitioning the Variation • The sum of squares due to factor Ais labeled SSA. It measures the squared differences between the mean of each level of factor A and the grand mean. • The sum of squares due to factor Bis labeled SSB. It measures the squared differences between the mean of each level of factor B and the grand mean.

  26. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • The sum of squares due to the interacting effect of A and Bis labeled SSAB. It measures the effect of combining factor A and factor B. • The sum of squares erroris labeled SSE. It measures the variability in the measurements within the groups. • Thus, the total variation is divided into four components: SST = SSA + SSB + SSAB + SSE

  27. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS Using the ANOVA Table in a Two-Way Design • The ANOVA table for such a design looks just like the ANOVA table for a one-way design with two additional rows.

  28. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS Using the ANOVA Table in a Two-Way Design • In a two-way ANOVA, three hypothesis tests should be done. • To test the hypothesis of no difference due to factor A we would have the following null and alternative hypotheses: Ho:There is no difference in the population means due to factor A. HA:There is a difference in the population means due to factor A.

  29. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • To test the hypothesis of no difference due to factor B we would have the following null and alternative hypotheses: Ho: There is no difference in the population means due to factor B. HA:There is a difference in the population means due to factor B.

  30. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • To test the hypothesis of no difference due to the interaction of factors A and B we would have the following null and alternative hypotheses: Ho:There is no difference in the population means due to the interaction of factors A and B. HA:There is a difference in the population means due to the interaction of factors A arid B.

  31. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS Understanding the interaction Effect • The easiest way to understand this effect is to look at a graph of the sample averages for each of the possible combinations of the two factors. • The line graph shown in Figure 14.7 displays the 20 sample means for airspace.

  32. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • From this graph you can see that the mean airspace decreases the longer the box sits on the shelf, regardless of from what position in the hardroll the box was made. • The airspace behavior is affected by the interaction of the time on the shelf and the position in the hardroll from which it was made.

  33. SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS • If there were no interaction effect, the lines connecting the sample means would be parallel as in Figure 14.8.

  34. SECTION 2MULTIPLE COMPARISON PROCEDURE • When we use analysis of variance to test whether the means of kpopulations are equal, rejection of the null hypothesis allows us to conclude only that the population means are not all equal. • In some cases we will want to go a step further and determine where the differences among means occur.

  35. SECTION 2MULTIPLE COMPARISON PROCEDURE • The purpose of this section is to introduce two multiple comparison procedures that can be used to conduct statistical comparisons between pairs of population means.

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