Business Statistics Chapter 13 One Way Anova

Fisher's LSD T-Test Version

1. Calculate critical value based on N-K degrees of freedom - k = categories 2. compare t value to the critical value

SST Calculation

1. SSTR + SSE 2.

Hypothesis Testing with the F

1. State hypothesis 2. Specify level of significance and determine the characteristics of comparison distribution 3. Calculate Test Statistic (using source table) 4. Determine the critical value on the comparison distribution and write a rule (Decision rule: If F(obt) >= Critical Value, then reject the null Conclusion Statement: F(df num, df denom) = F calculated, p</> alpha

Pairwise Null and Alternative Hypothesis

A null and alternative hypothesis exists for every pairwise comparison being done after the omnibus F has been rejected

Analysis of Variance (ANOVA)

A procedure capable of analyzing mean differences between more than two groups simultaneously Built-in Assumptions: - Scores must be independent - Populations are normally distributed - All populations have the same variance

Partitioning

Allocating the total sum of squares and degrees of freedom to the various components

Logic of Structural Model Approach Continued

Basic idea is that the distance between any individual's score (x) and the grand mean (x double bar) can be divided into two parts: - distance reflecting the variability that might exist due to any treatment effects (alpha j) - distance between group mean (x bar) and grand mean (x double bar) - distance reflecting variability in respondents who were exposed to the same treatment (e part) - distance between individual score (x) and group mean (x bar)

Concerns with Bonferroni Adjustment

Bonferroni allows us to make multiple comparisons after rejecting the omnibus F test - when a lot of multiple comparisons are to be made, this increases the likelihood of making a type II error If a lot of comparisons are to be made, using Tukey's HSD is better

Bonferroni Adjustment

Changes the overall experiment wise error rate from alpha(ew) to alpha(ew)/C

Post-hoc Comparisons

Computed only after the rejecting the null for omnibus F Many tests available - most involve looking at all possible pairwise comparisons among groups - meant to keep Type I error rate in check

Three Applications of Fisher's LSD

Df are N-K

In ANOVA treatments are

Different levels of a factor

Between Groups Estimate of Variation

Due to group membership called Mean Square due to Treatments (MSTR)

Logic of MSE and MSTR

If null = true - MSE and MSTR would reflect the same variance - ratio of MSE: MSTR would be 1:1 If null = false - MSTR also includes any differences based on treatment or group differences - ratio of between to the within will be greater than 1:1 MSTR/MSE = F Ratio

Harmonic Mean

If sample size per group is unequal , you can calculate a mean that allows you to use the Tukey's HSD n~ = Number of Groups / sum(1/ni) if three group experiment with sample sizes of 4,4,3 n~ = 3/(1/4 +1/4 + 1/3)

Confidence Intervals with Fisher's LSD

If the confidence interval contains 0, we claim that no difference exists

Structural Approach and the Null Hypothesis

If the null hypothesis is true, the division of the overall deviation into two parts should be random, resulting in population estimates that produce an F ratio of about 1. If the alternative hypothesis is true, the variance representing treatment effects should be greater than the variance representing sampling error, resulting in population variance estimates that produce an F ratio > 1.

Turkey 's HSD Formula

If your df isn't there, use the next most conservative value aka lower

Logic of Structural Approach

Individual Score Can be predicted in the following way:

Logic of the Structural Model Approach Part 3

Look at x, x(bar), and x(double bar) x double bar - x = SST x double - x bar = SSTR x bar - x = SSE

Analysis of Variance

Looks for mean difference through a comparison of variances - between group variance examines difference between the means of each group - within group variance examines how respondents in the same group differed from one another

Tukey's HSD

Maintains overall experiment alpha level, which means that even when a large number of pairwise comparisons are made, the integrity of the chosen Type I error rate is preserved

Basic Logic of Anova

Null Hypothesis: All groups are randomly drawn from identical populations - the population mean 1 = pop mean 2 = pop mean 3 Alternative Hypothesis: At least two groups are drawn from populations with different means - at least one population mean is different

Fisher's LSD w/o Bon Ferroni Adjustment

Only appropriate if there is one, single post-hoc comparison being performed

Fisher's Least Significant Difference

Reject Null if absolute value of (x bar a - x bar b) >= LSD

Pairwise Comparisons

Rejecting the null hypothesis of an omnibus F test doesn't tell us about which groups differ from one another = comparison of two group means to determine if they are statistically different from one another - common to perform multiple

Conclusion Statement

Results from our branding study suggested that the name of the manufacturer did influence desirability of the product, F(2,27)=4.09, p<.05. Fisher's LSD with Bonferroni adjustment as well as Tukey's HSD were used to probe the significant effect. Both post hoc tests showed that those who believed the phone was made by Samsung (x bar = 8) gave significantly higher desirability ratings than those who believed the phone was produced by Nokia (x bar = 4.5). No other significant differences emerged.

Conclusion of Structural Approach

SS Total = SS Error + SS Treatment MS(TR) = SS(TR)/df(TR) MS(E) = SS(E)/df(E) F = MS(TR)/MS(E)

Maintaining Experiment-Wide Alpha

Series of T-Tests would lead to alpha inflation - alpha (ew) = certain alpha level of the omnibus F Performing multiple t-tests would lead to the inflation of the alpha Post-hoc tests try to maintain alpha EW while maintaining adequate power to detect the effects

F Distribution

Shape dependent on the numerator and denominator degrees of freedom Numerator degrees of freedom are the degrees of freedom of the between-group variance estimate: df(tr) = k-1 --> k = number of treatments or groups Denominator DF = df error = N-k -- N = total observations k = treatment groups

Calculating SSE

To calculate the sum of squares due error (e.g., the within group variance), we need to determine how much variation is present in each group despite all people in a group getting the same treatment.

Calculating SSTR

To calculate the sum of squares due to treatments (e.g., the between group variance), we need to determine how far away each group mean is from the grand mean. Then, we weigh each deviation by the sample size of that group (n=10). Grand mean (x double bar) is the mean of all observations regardless of group membership

Computing LSD for Multiple Comparisons: Bonferroni Adjustment

Use adjusted alpha to calculate the t value

Tukey's HSD results

Use the value calculated through Tukey's HSD in the same way you would LSD - compare it to the mean differences

Decision Rule

Using Bonferroni Adjustment - calculate the LSD and compare that to the absolute value of any mean difference - each mean difference must be greater than that to be rejected

Tukey's HSD: Q Values

Utilizes q values which come from the studentized range distribution q value chosen based on - k (number of treatment groups) - df(error) = df within - Chosen alpha level

Within Groups Estimate of Variation

Variation within each group is called the Mean Square Error (MSE)

Concerns with Fisher's LSD

We should perform post hoc tests with the intention of trying to keep experiment wise alpha (alpha ew) constant comparisons related to one omnibus F, so chance we commit type I error is actually much higher

ANOVA Procedure is

a statistical approach for determining whether or not the means of three or more populations are equal

In ANOVA factor refers to

the independent variable