Biostat exam 4
Use Kruskal-Wallis in real-world with
Dominance hierarchies and developmental stages
Use which post-hoc following a Kruskal-Wallis null hypothesis rejection
Dunn's test
Use Kruskal-Wallis to determine if ___ samples come from _____________ identical populations
k, identical
What are you interested in when performing a Tukey test?
Examining mean differences
Purpose of post hoc test
Exploratory. You test the difference between all potential combinations of groups
What is the test statistic for ANOVA?
F statistic
Which type of ANOVA when researchers want to compare anxiety levels of 6 individuals at two marital states: after divorce, and again after they have remarried? Anxiety is measured 3 times
Factorial ANOVA (2 dependent factors)
Which type of ANOVA when researchers want to test new anti-anxiety medication and they measure anxiety of 36 participants on 3 different dosages, and they are also divided on whether they are high school or college students?
Factorial ANOVA (2 independent variables)
Which type of ANOVA when researchers want to see if HS students or college students have different levels of anxiety as they progress throughout the semester?
Factorial ANOVA (2 mixed factors)
Dunnett's test is designed to hold the ____________ at or below alpha when performing multiple comparisons of treatment groups with a control.
Familywise error rate
the probability of a coming to at least one false conclusion in a series of hypothesis tests . In other words, it's the probability of making at least one Type I Error.
Familywise error rate
The different groups are interesting and you want to know which are different from each other
Fixed effect/model I anova
Nonparametric alternative to the repeated measures ANOVA
Friedman's test
The Friedman test null hypothesis
H0: The treatment effects have identical effects
The Friedman test alternative hypothesis
Ha: At least one treatment is different from at least one other treatment
In order to reject the null hypothesis that the group means are equal, we need a _____ F-value
High
Risk of type II error for a post hoc test
High Not unlikely to generate a false negative (accepting the null when it is false) because this test lacks power
Which post hoc method can be used to address the overly-conservative nature of the Bonferroni correction?
Holm method
Why is it not appropriate to use independent sample t-tests to test all possible pairs of treatments and to identify differences between treatments rather than an ANOVA?
If multiple t-tests are carried out, then the type I error rate will increase with the number of comparisons made, ANOVA avoids this problem
A one-way ANOVA can be used to determine what?
If the means of three or more groups are the same
Interested in how the variation among families compares to the variation between families in model II one way anova
Partitioning of variance
Which type of test has more power between post hoc and planned comparison tests?
Planned comparison tests
Although some independence is assumed for the Kruskal-Wallis test, the data does not need to be...
normally distributed
For Kruskal-Wallis, smallest rank gets what number?
1
Assumptions of Kruskal Wallis test
-Random samples from their respective populations Independence within each sample and mutual independence between samples The measurement scale is at least ordinal (data can be ranked)
Model II is looking mostly at
1 family and within that
Kruskal-Wallis assumptions
1) All samples are random from respective populations (use >5 samples/group) 2) Independence within each sample and mutual independence among various samples 3) Measurement scale is at least ordinal (data can be ranked) 4) Homoscedasticity
The Friedman test is based on the following assumptions:
1)The Friedman test assumes that there are k experimental treatments (k ≥ 2) The rows are mutually independent. That is, the results within one block (row) do not affect the results within other blocks. 3)The data can be meaningfully ranked. Note: Average ranks are used in the case of ties.
What is the null hypothesis of a nested ANOVA with two levels?
1. that the groups have the same mean 2. the subgroups within each group have the same means
A Bacteriologist wishes to compare the optimal growth temperature for the bacteria Escherichia coli and Salmonella that cause food poisoning. He studied 12 petri dishes of each bacterium and counted the amounts of spores formed at 35, 45 and 55 ℃ to see if there was a significant difference amongst the 3 null hypotheses. The alpha level was set to 0.05. Which test?
2-Way Independent Variables ANOVA
Why is the Bonferroni correction often considered too conservative?
Protects (sometimes too well) against type I error (rejecting null hypothesis when it's actually true)
Null hypothesis of the one way ANOVA
All of the groups (means) are equal
Which test to determine if k samples come from identical populations?
Kruskal Wallis test
Dunn's test is frequently used with
multiple comparison adjustments
Benjamini and Hochberg reasoned that:
α should be interpreted as a desired false discovery rate (FDR) and should reflect how the expected rate of false discoveries changes after some pairwise tests are rejected in sequence
A Researcher wishes to compare the amount of blood glucose in 50 randomly chosen Diabetic individuals labeled 1-50 at 2 stages: before and after starting a new insulin treatment. The amounts of blood glucose levels were tested at 2, 4, and 6 weeks before and after the study to see if there was a significant difference amongst the 3 null hypotheses. The alpha level was set to 0.05. Which type of test is appropriate?
2-way dependent variables ANOVA
In an ANOVA, the observed variance of a variable is partitioned into what?
components attributable to different sources of variation
Allows us to evaluate the multiple layers and hierarchies that exist in datasets
ANOVA (analysis of variance)
An adult human has 32 teeth. However, as we get older we face events that can cause us to lose our adult teeth. Examples include losing a tooth from an accident, getting a tooth pulled due to infection or crowding, and having your wisdom teeth removed. You went to five different universities across Texas and randomly sampled twenty students from each university to provide you with how many adult teeth they still have. You are excited to analyze your data and you have decided to do a One-Way ANOVA. Questions: What is your Among Groups Degrees of Freedom and what is your Within Groups Degrees of Freedom? If you reject your null hypothesis with the One-Way ANOVA, should you perform a Tukey-Kramer Test, a Bonferroni Test, or a Dunnett's Test?
Among Groups DF=4. Within Groups DF=95. Tukey-Kramer Test.
What is the question evaluated by ANOVA?
Are the sample means different across the samples?
When looking at running speed before, one week into, and three weeks into a program of exercise, why would you use a two way ANOVA over a one way ANOVA?
Because individuals would start with different running speeds
Which test is commonly used following ANOVA evaluating medical experiments, for example comparing blood count measurements on three groups of animals, one of which served as a control while the other two were treated with two different drugs?
Dunnett's test
What does an ANOVA do?
Compares means by detecting differences in variance between the 3 or more groups under investigation
Purpose of the planned comparison tests
Confirmation theory or hypothesis. You test only the groups you expect to be different in a specific direction (i.e. greater, lower)
Model I is looking at
Different families and comparing them to each other
Homoscedasticity for Kruskal-Wallis
Different groups have the same distribution Groups with different standard deviations have different distributions
For the Kruskal-Wallis test, both groups' __________ should be about the same
Distribution Can't have one that's normal and one that isn't
Non parametric data
Does not assume data comes from a distribution that can be completely described by 2 parameters, mean, and standard deviation
In many research studies, a control is run along with the test groups and the question becomes: which groups differ from the control or baseline response. Which post-hoc test is most appropriate?
Dunnett's test
In which post-hoc test is the control identified and multiple comparisons are made to it?
Dunnett's test
Post hoc test for the Kruskal Wallis test
Dunns test
"within" group variation is often called what?
Error
Difference between one way ANOVA and a factorial ANOVA with independent factors
In the factorial, you're dealing with more than one independent variable
The repeated-measures ANOVA has one factor with at least 2 levels, which are
dependent
For the two factor ANOVA, the results of each of the three distinct tests are _______________ of the others.
Independent
The two-way ANOVA is used to analyze the effect of which variable?
Independent variable on the expected outcome, along with the relationship to the outcome itself
What happens to the F-value when when the variability of group means is large relative to the within group variability?
It is higher
What happens to the F-value when the group means are close together (low variability) relative to the variability within each group?
It is lower
Test for identical populations, designed to be sensitive to unequal means
Kruskal-Wallis
Which test do you convert measurement observations to ranks?
Kruskal-Wallis
Non-parametric test for whether the mean ranks are the same in all the groups
Kruskal-Wallis test
Use this test for a non-parametric version of ANOVA
Kruskal-Wallis test
Risk of Type I error for a planned comparison test.
Low. It is conservative, but not as conservative as a post hoc test
Risk of type II error for a planned comparison test
Lower than the post hoc test because it is more sensitive, and you are more likely to uncover a difference. It has more power
Three distinct tests of a two factor ANOVA
Main effect of Factor A Main effect of Factor B Interaction of A and B
Non-parametric analogue to 2 sample t test
Mann-Whitney U aka Wilcoxon rank sum
In the one way ANOVA, you make multiple observations of the __________ variable for each value of the _______ variable
Measurement Nominal
Which model for ANOVA is for fixed effects?
Model I
Partitioning of variance applied to which model of ANOVA?
Model II
Which model of ANOVA is for random effects?
Model II
Calculate df total for one way ANOVA
N-1
Calculate df within in one way ANOVA
N-a N= total # observations a= how many levels of factor you have
___________ accomplishes a partitioning of variance between treatment means, sampling group means, and the residuals.
Nested ANOVA
Which variable is nested in a nested ANOVA?
Nominal
Which variable is nested in the nested ANOVA?
Nominal
Mann-Whitney U aka Wilcoxon rank sum
Nominal variables with only 2 values
Kruskal-Wallis is a non-parametric test for
Non-normal ANOVA Best solution with one nominal variable and one ranked variable
A researcher wants to test the amount of antioxidants present in different types of tea. The types of tea that are tested are Green Tea, Black Tea, Herbal Tea, and Mint Tea. The amounts of antioxidants are measured in milligrams of antioxidant per gram of tea (mg/g). The amounts of antioxidants are measured multiple times for each type of tea, but not with the same bag (measurements are not made on the same individual tea bag more than once; different tea bags are used). The observations for amount of antioxidants in each type of tea fit a normal distribution and the distributions for each category of tea have the same standard deviation. With this information, and knowing that the observations were made independently, which statistical test should the researcher use to analyze their data? If the observations did not fit a normal distribution but all other information remained the same, what statistical test should the researcher perform then?
One-Way ANOVA for the first scenario; Kruskal-Wallis Test for the second scenario.
Degrees of freedom considered for the one way ANOVA
df between df within df total
Assumptions of two way ANOVA
Observations within each cell are normally distributed and have equal standard deviations
One way ANOVA assumptions
Observations within each group are normally distributed Data are homoscedastic
Variables of nested anova
One measurement variable and more than one nominal variable
Use Kruskal-Wallis with what types of variables?
One nominal and one measurement variable that is severely non-normal OR One nominal variable and one ranked variable
Test for analyzing the effect of three different diets on total serum cholesterol
One way ANOVA
A researcher has decided to test whether there is a difference in the means of weight in Asian, African and Caucasian students at an elementary school. The weights for 10 members of each population were acquired. Which type of ANOVA test should be performed? If you are determined to find where any differences are located, and you find a critical value of 2.93 and calculate an F statistic of 6.46, what post hoc test, if necessary, should be conducted?
One way ANOVA, Tukey's Test
Which test is used to determine whether there are any statistically significant differences between the means of three or more independent groups?
One-way ANOVA
The factorial ANOVA (with 2 mixed factors) is kind of like the combination of which two tests?
One-way ANOVA and repeated-measures ANOVA
Friedman test can be performed on what kind of data?
Ordinal (ranked) data
The repeated measures ANOVA is the extension of which test?
Paired t-test
The different groups are random samples from a larger set of groups and you're not interested in which groups are different from each other
Random effect/model II
In the ANOVA, if the F statistic is greater than the critical value, how do you treat the null hypothesis?
Reject it
A ____________ study is typically performed with a time series or treatment series (as independent variable).
Repeated measures
Which type of ANOVA for comparing 3 or more group means where the participants are the same in each group?
Repeated measures
Experimental design options for two way ANOVA
Repeated measures Randomized blocks
The Two Way/Factorial ANOVA (with 2 dependent factors) is like an extension of the ___________________, except you are dealing with 2 independent variables, rather than 1.
Repeated measures ANOVA
Which type of ANOVA for testing new anti-anxiety medication, measuring anxiety of 7 participants three times?
Repeated-measures ANOVA
Assumptions of ANOVA models
Samples are randomly selected from the populations and are independent from each other Response variable is normally distributed The means of the distributions are the same variance
Alternative hypothesis for Kruskal-Wallis
Samples came from populations with different distributions
Null hypothesis for Kruskal-Wallis
Samples came from populations with the same distribution
The repeated-measures ANOVA is almost identical to the one-way ANOVA, except for one additional calculation which accounts for what?
Shared variability between the levels
What is the Kruskal-Wallis test?
non-parametric test for the situation where the ANOVA normality assumptions may not apply
You are performing a pilot study on the reduction of serum triglycerides by a new pharmaceutical over a 6 week period. You have 8 volunteers (yes...all IRB approved and all!) who will take the prescribed dose daily for 6 weeks and you collect triglyceride levels (5 ranges) from their blood samples at the end of week 2, week 4 and week 6. What would be the most appropriate statistical test to evaluate whether there is any difference in the serum triglyceride levels in week 2, 4 & 6?
The Friedman test (equivalent to a repeated measures ANOVA and due to the small sample size and ordinal data collected this test is the most appropriate)
What does the F statistic compare?
The average variability between the groups to the average variability within the groups
Most important aspect when considering if you should use a Kruskal-Wallis test
The data is ordinal (ranked)
The shape of the F distribution of the one way ANOVA depends on which degrees of freedom?
df for among-group (between) variance df for within-group variance
Dunnett's test is well known and widely used in multiple comparison procedure for simultaneously comparing all active treatments with a control when _______________________ can be accepted.
The normality assumption for sampling can be accepted
The test statistic for a one-way ANOVA, F, represents what?
The ratio of variance among means divided by the average variance within groups
researchers surveyed two different majors with 6 individuals in each group: students studying biology and students studying chemistry. The experiment was conducted over a 6 week period in which students were told to run a minimum of 4 miles in a week's time. After each week, each student was told to rank his or her stress level from 1-10 (1 being the lowest, 10 being the highest). The researchers surveyed the same individuals after each week period, indicating that six studied biology and six studied chemistry. After the 6-week period, the research group ran a Mixed ANOVA to see if there was a significant difference in stress levels, but it was apparent that a mistake was made in their calculations. Which of the following statements could potentially be the reason the group made in error in this type of ANOVA? How could this mistake be fixed?
The research group calculated an error term for each separate week, indicating that more than two error terms were present in their calculations. This mistake could have been fixed if the research group only calculated an error term for the specific major and an error term for the interaction of each week. By calculating an error term for the interaction and for each week, unnecessary error terms were included which skewed the experiments results. (In their calculations, only two error terms are needed: one for the type of major and one for the interaction of the stress levels over the weeks. A third error term is unnecessary in this type of an ANOVA, so the research group should eliminate the extra error term used to calculate the final F-statistic.)
Null hypothesis of Kruskal Wallis test
The samples come from populations with equal medians OR The samples come from populations with the same distribution
Repeated measures ANOVA is derived from the 1-way ANOVA with what type of groups?
non-independent
Why would you use ANOVA over a 2-sample t-test?
There are more than 2 groups/samples
Why can repeated measures designs be very powerful?
They control for factors that cause variability between subjects
What is meant when the levels for an ANOVA are dependent?
They share variability in some way
If you reject the null of a one way ANOVA, which post hoc test should you use?
Tukey's test
Which post hoc test can tell the researcher which groups among the sample in specific have significant differences
Tukey's test
Variables for two factor ANOVA
Two or more independent variables (factors)
Researchers wanted to compare job satisfaction of new Walmart employees with new HEB employees over the course of their first four months. 5 newly hired employees at Walmart and 5 newly hired employees at HEB were asked to rate their job satisfaction on a scale of 1-10 after every month. The statistical results are provided: Month: F = 5.478 (critical value = 4.342)Walmart v. HEB: F = 2.365 (critical value = 2.675)Interaction: F = 4.389 (critical value = 3.678) Which choice best describes the most appropriate test statistic used as well as the conclusions reached from the data provided?
Two way ANOVA (mixed factors); New hires at Walmart and HEB did not have significantly different job satisfactions. There was a significant difference between the different months. There was an interaction effect present.
When you have rejected the null after the one way ANOVA, what would happen if you tried to do multiple comparisons on the data?
You increase the probability that at lease 1 will have a p value of less than 0.05 purely by chance, even if the null hypothesis of each is true
Dunn's test is a
non-parametric pairwise multiple comparison procedure
You are developing the data from a study in which you have measured enzyme activity in a gene product for which you have developed 2 new alleles by site directed mutagenesis that each altering an amino acid in the protein relative to the wild type. Since this protein is a dimer you now can have 6 possible genotypes observed in your randomly mating mouse population. You have randomly harvested 50 individuals from your population, divided them by sex and genotype, and measured enzyme activity in the brain of each mouse. What would the most appropriate analysis be that would help determine if the mean enzyme activity of the six genotypes are equal, the mean enzyme activity of the males and females are equal, and there is no interaction between the genotype and sex of the mouse affecting enzyme activity?
Two-way (factorial) ANOVA
The reason we cannot use multiple t-tests to claim that 4 populations have the same mean is that we increase the likelihood of which type of error?
Type I
Although the Kruskal-Wallis test is for identical populations, it is designed to also be sensitive to what?
Unequal means
Welch's ANOVA is a parametric test that is robust when it comes to what?
Unequal variances
Risk of Type I error for a post hoc test
Very low. Very unlikely to generate a false positive. Reject the null hypothesis when it is true.
What is accomplished by nested ANOVA?
a partitioning of the variance between treatment means, the sampling group means and the residuals (or individual variation between the subsamples and duplicate measures)
Calculate df between in one way ANOVA
a-1 a= how many levels of factor you have
Bonferroni adjustment leaves ________ numerically intact, but multiplies the ________
alpha, P value
A modification of hierarchical ANOVA developed to evaluate molecular genetic variance within and among demes of subdivided populations
analysis of molecular variance (AMOVA)
The Friedman test is a non-parametric test for
analyzing randomized complete block designs equivalent to a repeated measures ANOVA
In ANOVA, Variance ______ treatments needs to be greater than the variance ______ treatments so that the F value is high, in order to get a statistical difference and reject the null hypothesis
between, within
Why is it necessary to conduct post hoc comparisons between pairs of treatments after a one way ANOVA?
When a significant effect has been found using analysis of variance, we still do not know which means differ significantly
When would you use Welch's instead of a one way ANOVA?
When the data is heteroscedastic
The Tukey test is more liberal than the Scheffe test. What does this mean?
When you perform a Tukey test, you have a greater chance of rejecting the null hypothesis, and therefore making a type I error
What does the one way ANOVA test?
Whether the means of the measurement variable are the same for different groups
What does Dunns test help you find?
Which group is the different one
Wile ANOVA can tell the researcher whether groups in the sample differ, what can it not tell the researcher?
Which groups differ
The Tukey test is used after a one way ANOVA to determine what?
Which groups in the sample differ
You want to know whether or not high intensity and low intensity workouts have a difference in amount of weight lost compared to groups of males and females. You take a sample of 100 people 50 males and 50 females, then randomly assign them to an intensity level workout regime. This creates the groups A: High intensity males, B: low intensity males, C: high intensity females, D: low intensity females. At the end of the experiment each groups weight loss was averaged and compared to each other. The means were as followed Group A: 20 lbs, B: 12 lbs, C: 10 lbs, D: 22 lbs. Does this data reveal any main effects or an interaction?
Yes, there appears to be an interaction, but no main effect
Basic idea of the one way ANOVA
calculate the mean of the observations within each group, then compare the variance among these means to the average variance within each group
While Kruskal-Wallis does not assume that the data are normal, it does assume that the different groups have the same _________, and groups with different standard deviations have different distributions.
distribution
Difference between two way ANOVA and nested ANOVA
each grouping extends across the other grouping (each genotype contains some males and some females, and each sex contains all three genotypes)
The ________ sum of squares measures the variability of the observed values around their respective treatment means.
error
As the variability due to chance decreases, the value of F will
increase
The Bonferroni adjustment
introduced the familywise error rate (FWER)
Why is ANOVA especially well suited for biomedical research?
it can evaluate interaction between variables
A Tukey-Kramer Test calculates a ________ value, and if any difference between two group means exceeds this value, then these two groups are statistically different.
minimum significant difference (MSD)
Dunn's test uses a Bonferroni adjustment to
modify the rejection level for any test by dividing alpha/ total number of tests
Variables for nested ANOVA
one measurement variable and more than one nominal variable, and the nominal variables are nested (form subgroups within groups)
Variables for two way ANOVA
one measurement variable and two nominal variables, and each value of one nominal variable is found in combination with each value of the other nominal variable
Variables for one way ANOVA
one nominal variable, which divides the measurements into 2 or more groups one measurement variable
A Tukey-Kramer Test is specifically for ________ ANOVA post-hoc evaluation
one-way
Kruskal-Wallis test is the non-parametric alternative to which test?
one-way ANOVA
ANOVA is a _______ test
parametric
All of the ____________ assumptions apply for the two factor ANOVA.
parametric
Dunnett's test wastes less _________ than a Bonferroni approach
power
Test statistic for Tukey's test
q
Use Kruskal-Wallis when the measured variable is
ranked/does not meet normality assumptions
familywise error rate (FWER)
redefines α to signify the probability of falsely rejecting the null hypothesis in one test out of all tests performed
Between nested and repeated measure ANOVA, which assumes homogeneity of variance?
repeated measures
a natural extension of the paired t test when you have more than 2 datasets from the same individuals being compared
repeated measures ANOVA
For the one way ANOVA, it is not necessary to have equal what?
sample sizes
The Friedman test is an extension of the:
sign test when there may be more than two treatments.
Bonferroni adjustment requires a much __________ p value to reject null hypothesis
smaller
ANOVA behaves as a ______ on more than 2 groups
t-test
The test statistic F for ANOVA is equal to what?
the 'between treatments' mean square divided by the error (within) mean square
What does ANOVA allow us to analyze?
the differences between group means and deeper relationships, such as "variation" among and between groups
What does the two way ANOVA compare?
the mean differences between groups that have been split by two independent variables or factors
Null hypothesis of two way ANOVA
the means of the measurement variable are equal for different values of the first nominal variable; that the means are equal for different values of the second nominal variable; and that there is no interaction (the effects of one nominal variable don't depend on the value of the other nominal variable)
In a nested ANOVA, all of the lower level subgroupings must be random effects (model II) variables, what does this mean?
they are random samples of a larger set of possible subgroups
Purpose of Tukey's test
to determine which groups in the sample differ
The primary interest of designing a randomized block experiment
to reduce the within-treatments variation to more easily detect differences among the treatment means
Holm's sequential adjustment; and the Holms-Sikak sequential adjustment
treats subsequent pairwise hypothesis tests as parts of different families on the basis of whether previous tests were rejected
Which test to determine if there is interaction between the two independent variables on the dependent variable
two way ANOVA
Which test for determining if there is an interaction between sex and alcohol consumption on athletic performance, where sex (male & female) and Alcohol consumption (number beverages per week) are the independent variables, and athletic performance (measured by length of time on elliptical runner at a set rotations/min) is your dependent variable
two-way ANOVA
Calculate F
variance between treatments _______________________ variance within treatments
When should you use the Kruskal-Wallis test?
when you have one nominal variable and one measurement variable that is severely non-normal, or when you have one nominal variable and one ranked variable
In its basic design, ANOVA can test what?
whether or not the means of several groups are not significantly different
What does the interaction term in a two way ANOVA tell you?
whether the effect of one of your independent variables on the dependent variable is the same for all values of your other independent variable (and vice versa)
What does the nested ANOVA test for?
whether there is significant variation in means among groups, among subgroups within groups, etc
The Kruskal-Wallis test can be less powerful than one-way ANOVA because
you lose information when you sub ranks for original values