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1 11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview Overview of ANOVA Analysis of variance (ANOVA) is a comparison of means. ANOVA allows you to compare more than two means simultaneously. Proper experimental design efficiently uses limited data to draw the strongest possible inferences. Overview of ANOVA The Goal: Explaining Variation ANOVA seeks to identify sources of variation in a numerical dependent variable Y (the response variable). Variation in Y about its mean is explained by one or more categorical independent variables (the factors) ) or is unexplained (random error).
2 Overview of ANOVA The Goal: Explaining Variation Each possible value of a factor or combination of factors is a treatment. We test to see if each factor has a significant effect on Y using (for example) the hypotheses: H 0 : μ 1 = μ 2 = μ 3 = μ 4 H 1 : Not all the means are equal 1 q The test uses the F distribution. If we cannot reject H 0, we conclude that t observations within each treatment have a common mean μ. Overview of ANOVA Overview of ANOVA Overview of ANOVA The Goal: Explaining Variation For example, a one-factor ANOVA would test the hypothesis that the length of hospital stay (LOS) is affected by Type of Fracture: Length of stay = f(type of fracture) A two-factor ANOVA would test the hypothesis that t the length of hospital stay (LOS) is affected by Type of Fracture and Age Group: Length of stay = f(type of fracture, age group) We can also test for interaction between factors. The Goal: Explaining Variation
3 Overview of ANOVA ANOVA Calculations Software (e.g., Excel, MegaStat, MINITAB, SPSS) is used to analyze data. Large samples increase the power of the test, but power also depends on the degree of variation in Y. Lowest power would be in a small sample with high variation in Y. Overview of ANOVA ANOVA Assumptions Analysis of Variance assumes that the - observations on Y are independent, - populations being sampled are normal, - populations being sampled have equal variances. ANOVA is somewhat robust to departures from normality and equal variance assumptions. Data Format A one-factor ANOVA only compares the means of c groups (treatments or factor levels). Consider the format for a one-factor ANOVA with c treatments, denoted A 1, A 2,, A c Data Format Sample sizes within each treatment t t do not need to be equal (i.e., balanced). The total number of observations is equal to n = n 1 + n n c Hypothesis to Be Tested H 0 : μ 1 = μ 2 = = μ c H 1 : Not all the means are equal ANOVA tests all means simultaneously and so does not inflate the type I error.
4 as a Linear Model An equivalent way to express the one-factor model is to say that treatment j came from a population with a common mean (μ) plus a treatment effect (A j ) plus random error (ε ij ): y ij = μ + A j + ε ij j = 1, 2,, c and i = 1, 2,, n Random error is assumed to be normally distributed with zero mean and the same variance for all treatments. as a Linear Model A fixed effects model only looks at what happens to the response for particular levels of the factor. H 0 : A 1 = A 2 = = A c = 0 H 1 : Not all A j are zero If the H 0 is true, then the ANOVA model collapses to y ij = μ + ε ij Group Means The mean of each group is calculated as The overall sample mean (grand mean) can be calculated as Partitioned Sum of Squares For a given observation y ij, the following relationship must hold (y ij ij y ) = (y j y ) + (y ij y j ) Where (y ij y ) = deviation of an observation from the grand mean (y j y ) = deviation of the column mean from the grand mean (between treatments) (y ij y j ) = deviation of the observation from its own column mean (within treatments).
5 Partitioned Sum of Squares This relationship is true for sums of squared deviations, yielding partitioned sum of squares: Simply pyp put, SST = SSA + SSE Partitioned Sum of Squares SSA and SSE are used dt to test tth the hypothesis of equal treatment means by dividing each sum of squares by it degrees of freedom to adjust for group size. These ratios are called Mean Squares (MSA and MSE). The resulting test statistic is F = MSA/MSE. Partitioned Sum of Squares Partitioned Sum of Squares Use Excel s one-factor ANOVA (Tools > Data Analysis) to analyze data.
6 Test Statistic The F distribution describes the ratio of two variances. The F statistic is the ratio of the variance due to treatments (MSA MSA) ) to the variance due to error (MSE). Test Statistic When F is near zero, then there is little difference among treatments and we would not expect to reject the hypothesis of equal treatment means. Decision Rule F cannot be negative e has no upper limit. For ANOVA, the F test is a right-tailed tailed test. Use Appendix F or Excel to obtain the critical value of F for a given α. Decision Rule Steps in Performing Step 1: State the hypotheses H 0 : μ 1 = μ 2 = = μ c H 1 : Not all the means are equal Step 2: State the decision rule The degrees of freedom for the critical value F are Numerator d.f. = c 1 (between treatments) Denominator d.f. = n c (within treatments) For the given α,, obtain the critical value from Appendix F or Excel.
7 Steps in Performing For example, Steps in Performing Step 3: Perform the Calculations Use Excel to perform the calculations. For example, here are the results of an ANOVA: Steps in Performing Step 4: Make the Decision Reject H 0 if the critical value F exceeds the test statistic F α or if the p-value given by Excel is < α. MegaStat and MINITAB can also be used to perform the ANOVA. Multiple Comparison Tests Tukey s Test After rejecting the hypothesis of equal mean, we naturally want to know Which means differ significantly? In order to maintain the desired overall probability of type I error, a simultaneous confidence interval for the difference of means must be obtained. For c groups, there are c(c 1) distinct pairs of means to be compared. These types of comparisons are called Multiple Comparison Tests.
8 Multiple Comparison Tests Tukey s Test Tukey s studentized range test (or HSD for honestly significant difference test) is a multiple comparison test that has good power and is widely used. Named for statistician John Wilder Tukey ( ) This test is not available in Excel s Tools > Data Analysis but is available in MegaStat. Multiple Comparison Tests Tukey s Test Tukey s is a two-tailed tailed test for equality of paired means from c groups compared simultaneously. The hypotheses are: H 0 : μ j = μ k H 1 : μ j μ k Multiple Comparison Tests Multiple Comparison Tests Tukey s Test The decision rule is: Reject H 0 if y j y k MSE n j n k > T α Tukey s Test For example, here is the upper 5% of studentized range: Where T α = 0.707q c,n-c and q c,n-c is a critical value of the studentized range for the desired α.
9 Tests for Homogeneity of Variances ANOVA Assumptions ANOVA assumes that observations on the response variable are from normally distributed populations that have the same variance. The one-factor ANOVA test is only slightly affected by inequality of variance when group sizes are equal. Test this assumption of homogeneous variances, using Hartley s F max Test. Tests for Homogeneity of Variances Hartley s F max Test Named for H.O. Hartley ( ). The hypotheses are H 0 : σ 12 = σ 22 = = σ 2 c H 2 1 : Not all the σ j2 are equal The test statistic is the ratio of the largest sample variance to the smallest sample variance F max 2 max s 2 min max = s max Tests for Homogeneity of Variances Hartley s F max Test Assuming equal group sizes, critical values of F max are found using degrees of freedom Numerator df 1 = c Denominator df 2 = n/c - 1 Tests for Homogeneity of Variances Levene s Test Levene s test is a more robust alternative to Hartley s F max test. Levene s test does not assume a normal distribution. It is based on the distances of the observations from their sample medians rather than their sample means. A computer program (e.g., MINITAB) is needed to perform this test.
10 Two-Factor ANOVA Without Data Format In a two-factor ANOVA without replication (nonrepeated measures design) each factor is observed exactly once. The data are represented in an r rows by c columns matrix. The mean of Y can be computed either across the rows or down the columns. The grand mean y is the sum of all data values divided id d by the sample size rc. Two-Factor ANOVA Without Data Format Two-Factor ANOVA Without Two-Factor ANOVA Model Expressed in linear form: y jk = μ + A j + B k + ε jk Where y jk = observed data value in row j and column k μ = common mean for all treatments A j = effect of row factor A (j = 1, 2,, r) B k = effect of column factor B (k = 1, 2,, c) ε jk jk = random error (normally distributed, zero mean, same variance for all treatments) t t Two-Factor ANOVA Without Hypotheses to Be Tested Fixed-effects effects model: Factor A (row factor effect) H 0 : A 1 = A 2 = = A r = 0 H 1 : Not all the A j are equal to zero 1 j q Factor B (column factor effect) H 0 : B 1 = B 2 = = B c = 0 H 1 : Not all the B k are equal to zero If we fail to reject either H 0, then the model reduces to: y jk = μ + ε jk
11 Two-Factor ANOVA Without Randomized Block Model In this model, only one factor is of interest and the other factor is used to reduce variance. The column effects are treatments (the variable of interest). The row effects are called blocks. Subjects within each block are randomly assigned to the treatments. Computations are the same as a two-factor ANOVA but interpretation resembles a one-factor ANOVA. Two-Factor ANOVA Without Format of Calculations of Nonreplicated Two-Factor ANOVA Two-Factor ANOVA Without Randomized Block Model For example, let four types of fertilizer be the treatment and three soil types be the block. The data would be arranged as follows: Two-Factor ANOVA Without Steps in Performing Two-Factor ANOVA Step 1: State the hypotheses Factor A (row factor effect) H 0 : A 1 = A 2 = A 3 = 0 H 1 : Not all the A j are equal to zero Factor B (column factor effect) H 0 : B 1 = B 2 = B 3 = B 4 = 0 H 1 : Not all the B k are equal to zero 1 k q
12 Two-Factor ANOVA Without Steps in Performing Two-Factor ANOVA Step 2: State the decision rule The degrees of freedom for the critical values F are Factor A: : df 1 = r 1 Factor B: : df 1 = c 1 Error: df 2 =(r 1)( )(c 1) Two-Factor ANOVA Without Steps in Performing Two-Factor ANOVA Step 3: Perform the Calculations Use Excel to perform the calculations. For example, For the given α and appropriate degrees of freedom, obtain the critical values from Appendix F or Excel. Two-Factor ANOVA Without Steps in Performing Two-Factor ANOVA Step 4: Make the Decision Reject the null hypothesis if the F test exceeds the critical value or if the p-value is < α. Two-Factor ANOVA Without Multiple Comparisons You can use Tukey s simultaneous comparisons of the treatment pairs using a pooled variance. For example,
13 What Does Replication Accomplish? In this model, each factor is observed m times, with an equal number of observations in each cell (balanced data). Now we can test the factors (main effects) and also an interaction effect. This model is called a Full Factorial model. Format of Hypotheses Factor A: : Row Effect H 0 : A 1 = A 2 = = A r = 0 H 1 : Not all the A j are equal to zero Factor B: : Column Effect H 0 : B 1 = B 2 = = B c = 0 H 1 : Not all the B k are equal to zero 1 k q Interaction Effect H 0 :AlltheAB AB jk are equal to zero H 1 : Not all AB jk are equal to zero What Does Replication Accomplish? The linear form is y ijk ijk = μ + A j + B k + AB jk + ε ijk Where y ijk = observation i for row j and column k (i = 1, 2,, m) μ = common mean for all treatments A j = effect of row factor A (j = 1, 2,, r) ) B k = effect of column factor B (k = 1, 2,, c) AB jk = effect attributed t to interaction ti between factors A and B ε jk jk = random error (normally distributed, zero mean, same variance for all treatments) Format of Data
14 Sources of Variation Steps in Performing Two-Factor ANOVA Step 2: State the decision rule The degrees of freedom for the critical values F are Factor A: df 1 = r 1 Factor B: df 1 = c 1 Interaction (AB)) = df 1 = (r 1)( )(c 1) Error: df 2 = rc(m 1) Steps in Performing Two-Factor ANOVA Step 1: State the hypotheses Factor A: : Row Effect H 0 : A 1 = A 2 =, = A r = 0 H 1 : Not all the A j are equal to zero Factor B: : Column Effect H 0 : B 1 = B 2 = = B c = 0 H 1 : Not all the B k are equal to zero 1 k q Interaction Effect H 0 : All the AB jk are equal to zero H 1 : Not all AB jk are equal to zero Steps in Performing Two-Factor ANOVA Step 3: Perform the Calculations Use Excel to perform the calculations. For example, For the given α and appropriate degrees of freedom, obtain the critical values from Appendix F or Excel.
15 Steps in Performing Two-Factor ANOVA Step 4: Make the Decision Reject the null hypothesis if the F test exceeds the critical value or if the p-value is < α. Interaction Effect To visualize an interaction, plot the treatment means for one factor against the levels of the other factor. Connect the means within each factor. If no interaction, lines will be roughly parallel. If strong interaction, lines will have differing slopes and tend to cross each other. Possible Interaction Patterns Possible Interaction Patterns
16 Tukey Tests of Pairs of Means MegaStat performs Tukey comparisons. For example: General Linear Model What is the GLM? The general linear model (GLM) - is a tool for estimating large and complex ANOVA models - allows more than two factors - permits unbalanced data and any desired subset of interactions - provides predictions and identifies unusual observations - does not require equal variances - is easy to understand General Linear Model Higher-Order ANOVA Models ANOVA models can have more than two factors. For example: Interaction effects are AB, AC, BC, and ABC. Use MINITAB, SPSS, or SAS for higher-order models. Experimental Design: An Overview What is Experimental Design? Experimental design refers to - the number of factors under investigation, - the number of levels assigned to each factor, - the way factor levels are defined, and - the way observations are obtained. Fully crossed or full factorial designs include all possible combinations of factor levels. Fractional factorial designs limit data collection to a subset of possible factor combinations.
17 Experimental Design: An Overview What is Experimental Design? Nested or hierarchical designs occur when all the levels of one factor are fully contained in another. Balanced designs have an equal number of observations for each factor combination. Fixed-effectseffects models have predetermined levels of each factor, limiting inferences to only the specified factor levels. Random effects models randomly choose factor levels from a population of potential factor levels. 2 k Models Experimental Design: An Overview In a 2 k factorial design,, there are k factors, each with two levels. This reduces the data requirements in a replicated experiment. Especially useful when the number of factors is large. Experimental Design: An Overview Fractional Factorial Designs This design limits collection to a subset of the possible factor combinations for reasons of economy. Extremely important in real-life life situations where many factors exist. This model sacrifices some of the interaction effects. Experimental Design: An Overview Nested or Hierarchical Design All levels of one factor are fully contained within another. For example, Defects = f(experience, Method(Machine Machine)) Machine is nested within Method so the effect of Machine cannot appear as a main effect. Machine depends on Method.
18 Experimental Design: An Overview Random Effects Model Factor levels are chosen randomly from a population p of potential factor levels. Computation and interpretation is more complicated. Applied Statistics in Business and Economics End of Chapter 11
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