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