Univariate ANOVAs and Nonparametrics Flashcards
what are the different ANOVAs, univariable?
one way ANOVA
repeated measures ANOVA (single factor)
what are the different nonparametric tests of ANOVAs?
Kruskal-Wallis test
Friedman test
how do we compare data from more than 2 groups?
ANOVA
if we have three groups: people with Associates, Bachlors and no degree, what would our ANOVA test hypothesis be if we expect different mean scores among the groups?
mu1=mean of group 1 (Associates
mu2=mean of group 2 (Bachelors)
mu3=mean of group 3 (no degree)
H0: mu1=mu2=mu3
Ha: some mui are different
should we conduct multiple independent samples t tests for every pairing of groups if we have a sample with more than 2 groups?
no!
what are the problems with using multiple independent t tests for a sample with more than 2 groups?
this won’t directly evaluate the H0 vs Ha, just bits and pieces of it
this becomes more complex with larger #s of groups (or with 10 groups, there are 45 pairings)
compounding type 1 error (BIGGEST PROBLEM)
what is the biggest problem with running multiple independent t tests with a sample of more than 2 groups?
compounding type 1 error
what is an example of compounding type 1 error?
when you have 3 tests with 5% of type 1 error, this becomes a 14% chance of type 1 error in any of the three
what is family wise error rate (FWER)?
the probability of at least one type 1 error on a group of tests (the compounding type 1 error)
t/f: FWER becomes worse with more pairings
true
when we do multiple independent t tests for multiple groups, we want to test if Mu1=mu2=mu3…, but what we end up testing is:
mu1=mu2
mu1=mu3
mu2=mu3
t/f: in order to test H0 of more than 2 groups, pair off groups and conduct several independent samples t tests
false!!! do not do this!!!
what is the method for controlling FWER?
ANOVA
what is analysis of variance (ANOVA)?
the general procedure for making statistical comparisons of many types based on “breaking down” simple variance calculations
what is the numerator of the equation for sample variance?
sum of (xi-x bar) squared
the numerator of the equation for sample variance is equal to what?
sums of squares
calculating the sum of squares around a single, overall mean makes sense for one group, what is a better measure of overall dispersion for groups w/potentially dif means?
doing two “pieces” in sum of squares based around the group means and global/grand mean
what are the 2 pieces in sum of squares for multiple groups?
systematic variation
random variation
what is systematic variation?
differences among groups (deviations from the grand/global mean to group mean)
what is random variation?
natural randomness (deviation of data from the group means)
t/f: there are 2 sources of deviation from the overall mean
true
what are the 2 sources of deviation from the overall mean?
systematic deviation
random deviation
what is systematic deviation?
group averages to the overall average
what is random deviation?
observations (data points) to group averages
what are the 2 pieces of the sum of squares doing?
comparing the means among groups by comparing systematic and random deviations
is systematic deviation variation bw or w/in groups?
bw groups
is random deviation variation bw or w/in groups?
w/in groups
if systematic deviation is big…
group means may be dif
if random deviation is big…
lots of random error may be occuring w/in groups
if F statistic is large, the p is ___ and we ___ the null
small, reject
what are the formal names for the 3 deviations?
total sum of squares
bw groups sum of squares
w/in groups sum of squares
what is the total sum of squares (SST)?
deviations from observations to overall mean
sum of (observation-overall mean)^2
what is the bw groups sum of squares (SSB)?
deviations from group means to overall mean
sum of (group mean-overall mean)^2
when the groups means are different, is the SSB large or small?
large
what is the w/in groups sum of squares (SSW)?
deviations from observations to group means
sum of (observation-group mean)^2
when data are highly variable w/in a group, is the SSW large or small?
large
what is the equation for SST
SST=SSB+SSW
SST is the sum of what two things?
SSB and SSW
if the SSB is a larger contribute to SST than SSW, there is evidence of…
different groups means (Ha supported)
if SSB is the larger contributer to SST, if Ha or H0 supported?
Ha
if SSW is a larger contributer to SST than SSB, there is evidence of…
equal group means (H0 supported)
if SSW is the larger contributer to SST, is Ha or H0 supported?
H0
does SSB have a high inter-error deviation or intra-error deviation?
inter-error deviation
does SSW have a high inter-error deviation or intra-error deviation?
intra-error deviation
when data is clustered very close to the group means, is the SSB/SSW small or large?
SSW is small
if data is very dispersed around group means, is the SSB/SSW small or large?
SSW is large
if group means are dispersed from the grand mean, is the SSB/SSW small or large?
SSB is large
if group means are close to the grand mean, is the SSB/SSW small or large?
SSB is small
to conduct a hypothesis test for a one-way ANOVA, we need what two things?
a statistic based on SSB and SSW
a reference distribution for the statistic
t/f: we need the mean of squares BW groups and W/IN groups for one-way ANOVA
true
what is the MSB (mean of squares bw groups)?
MSB=SSW/(k-1)
what is k in ANOVA testing?
the # of groups (in the GRE example, k=3 for the Associates, Bachleors, and no degree groups)
what is the MSW (mean of squares w/in group)?
MSW=SSW/(overall sample size(n)-k)
t/f: ANOVA is an expansion of independent samples t test
true
what are two other names for the one way ANOVA test?
Fisher’s ANOVA
univariate ANOVA
what is the goal of the one-way ANOVA test?
to examine whether mean values from several (>2) populations are equal
what is the test statistic for one way ANOVA?
F
what is the F equation?
F=MSB/MSW
is the F distribution symmetrical?
nope
how many DF are required for the F-distribution?
2
what are the 2 DF used in the F-distribution?
MSB: k-1 DF
MSW: n-k DF
if the DF get really large, can it approach the z distribution standard normal curve appearance?
yes
what is an outcome in the one-way ANOVA?
the variable being analyzed (Y)
the dependent variable (DV)
what is a factor in a one-way ANOVA?
a characteristic that distinguishes groups
what are some examples of factors?
biological sex, treatment types, insurance status
what are levels in one-way ANOVA?
the groups defined by a factor
what are examples of levels?
male, female, and intersex in the factor of biological sex
intensive overground gait training, treadmill training, and exoskeleton training in the factor of treatment type
what are the assumptions for ANOVA?
continuous data
independence of data
normality
homogeneity of variance
how do we test for normality in ANOVA?
the SW test if n<30
how do we test homogeneity of variance in ANOVA?
Levene/s test
t/f: normality has a large effect on ANOVA
false, it has a small effect on ANOVA
t/f: homogeneity of variance has a greater impact on ANOVA results
true
the assumptions for ANOVA are the same as for independent samples t tests except what?
that ANOVA is still run if variance is not equal
the unknown group means are denoted as what in one way ANOVA hypothesis?
mui
are there one and two sided hypotheses with ANOVA testing?
no, just two sided
when the homogeneity of variance assumption of ANOVA is violated, what test is run?
Welch’s ANOVA
in one-way ANOVA, if p is less than or equal to alpha, do we support H0 or Ha?
Ha
when we support Ha, what has to be done? why?
further testing (pairwise comparisons) has to be run bc we this conclusion is nonspecific so we don’t know what groups are different just than some groups are different
what is the nonspecific conclusion of Ha?
there are differences in group means, but where the difference lies is unknown
when would we run pairwise testing in ANOVA?
when we support Ha
a study has 4 groups for teaching multiplication to elementary school children: (1) traditional tables, (2) word problem based, (3)integrated, (4) exploration based.
20 children are randomly assigned to each group and are taught multiplication for 2 months. At the end of the 2 months, a standarized multiplication test was administered to all children.
determine the effectiveness differences across the 4 teaching methods (alpha=0.05)
what is the DV?
what is the IV?
what is the factor?
what are the levels?
what kind of variable is this?
what is H0? Ha?
DV=test scores
IV=teaching methods
factor=teaching methods
levels=tables, word based, integrated, and exploration based
ratio variable
H0: mu1=mu2=mu3=mu4
Ha: some mui are different
how do we test the normality assumption for one way ANOVA?
SW test
what are the hypothesis for SW in one way ANOVA?
H0=normal
Ha: not normal
if the SW p value is greater than alpha, is this normal data? parametric or nonparametric?
data is normal and parametric (ANOVA)
how do we test the homoscedasticity in ANOVA?
Levene’s test
what are the hypotheses of the Levene’s test in ANOVA?
H0: equal
Ha: not equal
in the Levene’s test, if p>alpha, is this equal or not equal variance?
equal
in the Levene’s test, if p<alpha, is this equal or not equal variance?
not equal
the Levene’s test for ANOVA will give us data for “based on mean”, based on median”, “based on median with adjusted DF”, and “based on trimmed mean”, which should be used when data is normal?
“based on mean”
what is the test statistic for ANOVA?
F
t/f: the F statistic needs 2 DFs
true
if the teaching methods example has DFs of 3 and 76, test statistic of 48.109, and significance of <0.001, what is our conclusion?
the groups of elementary school children had significantly different mean multiplication test scores depending on the teaching method (F(3, 76)=48.109, p<0.001)
if we conclude from ANOVA that there is significant difference (p<alpha), what do we have to do?
run post hoc pairwise comparisons
if we conclude from ANOVA that there is no significant difference (p>alpha), do we have to run further tests?
nope
t/f: post hoc pairwise comparison is a way of controlling FWER
true
what are the dif methods of post hoc pairwise comparisons?
Tukey’s Honestly Significant Difference (HSD)
Bonferroni Correction
Holm Procedure
what does Tukey’s Honestly Significant Difference(HSD) do?
controls type 1 error by introducing a new distribution
best when all possible pairings of groups are tested
what is the Bonferroni correction?
a general method for type 1 error control
what does the bonferroni correction do?
adjusts the significance level (alpha)/type 1 error
hw do we do the bonferroni correction?
alpha/# of pairwise comparisons
suppose we want to conduct m multiple comparisons after a one way ANOVA, does m t-tests at level alpha have FWER no more than m x alpha or no more than alpha?
FWER no more than m x alpha
suppose we want to conduct m multiple comparisons after a one way ANOVA, does m t-tests at level alpha/m have FWER no more than m x alpha or no more than alpha?
FWER no more than alpha
if we have 3 comparisons and alpha is 0.05, what will our new alpha be with the bonferroni correction?
0.05/3=0.0166
after adjusting the significance level with the bonferroni correction, if any p value is less than the new alpha, is this significant?
yes
if you have 3 groups, how many comparisons would you have?
3
if you have 4 groups, how many comparisons would you have?
6
if you have 5 groups, how many comparisons would you have?
10
how would you report one way ANOVA results?
a one-way ANOVA was performed to compare the effect of [IV] on [DV] for [population]. results indicated that there [was/was not] a statistically significant difference in [DV] bw at least 2 groups (F(bw groups DF, w/in group DF)=[f-value], p=[p-value])
post hoc pairwise comparisons indicated that the mean value of [DV] was significantly different bw [group name] and [group name] (p=[p-value])
there was no statistically significant difference bw [group name] and [group name] (p=[p-value])
what is the independent samples nonparametric test for more than 2 samples?
Kruskal Wallis test
when do we run the Kruskal Wallis test?
when there is a categorical dependent variable
when the normality assumption for one-way ANOVA test is violated
what test extends the Mann Whitney U test (2 independent samples nonparametric test)?
Kruskal Wallis test
t/f: violation of the homogeneity of variance assumption of one-way ANOVA is a reason to run nonparametrics (Kruskal Wallis)
false, you would run the Welch’s ANOVA
what are the hypothesis of the Kruskal Wallis test?
H0: the populations have equal medians
Ha: the populations have unequal medians
does the Kruskal Wallis test have any one sided hypothesis?
no
what is the procedure for the Kruskal-Wallis test?
1) pool data from k groups into one data set
2) rank observations in the pooled data set
3) add up the ranks for every group (Rj)
4) calculate test statistic H
5) calculate p-value from chi squared w DF k-1)
what is the test statistic for the Kruskal Wallis test?
H
you have 30 pts w/ulnar fxs in the peds ER that are asked to rate their pain on a VAS of 1-20. 10 are randomly selected to receive a stuffed animal, 10 watched TV, and 10 got nothing. conduct a test on the data to determine if the median pain scores for these groups were equal.
stuffed animal: 9,8,4,12,14,9,14,6, 5,7
TV: 4,8,15,19,12,9,6,5,8,17
nothing: 6,5,19,8,10,13,20,12,12,15
ranks
stuffed animal: 14,10.5,1.5,18.5, 22.5,14,22.5,6,3.5,8
TV: 1.5,10.5,25,28.5,18.5, 14,6,3.5,10.5,27
nothing: 6,25,28.5,10.5,16,21,30, 18.5,18.5,25
H=1.999
DF=2
p=.368
what are the hypothesis?
what are the rank sums?
what test is being run?
what is the test statistic?
what is the conclusion?
Ho: m1=m2=m3
Ha: some medians are different
the Kruskal Wallis test is being run
R1=121
R2=145
R3=199
test statistic=H=1.999
p>alpha=fail to reject null
conclusion: the median scores are not significantly different among the different groups/the median scores are equal among the different groups
if there is large difference in the ranks sums, is it likely that we will reject or fail to reject H0?
reject H0 (support Ha)
if there is small difference in the ranks sums, is it likely that we will reject or fail to reject H0?
fail to reject H0 (support H0)
if the p from a kruskal Wallis test is less than alpha, do we run post hoc comparisons?
yes
if the p from a kruskal Wallis test is greater than alpha, do we run post hoc comparisons?
no
how do we run post hoc comparisons with nonparametrics (kruskal Wallis)?
the Mann Whitney U test
how do we run the post hoc test comparisons with parametrics?
2 independent samples t test
t/f: always run Bonferroni corrections for both parametric and nonparametric post hoc comparisons
true
when testing more than 2 groups, if any group n<30, what do we do?
run the SW test, call this p pn
t/f: always run the Levene’s test for multiple (more than 2) independent samples
true
if the Levene’s test p (pv) is less than or equal to alpha, are the variances equal or unequal?
unequal
if the Levene’s test p (pv) is greater than alpha, are the variances equal or unequal?
equal
if all the p values of the SW and Levene’s tests are greater than alpha, what test should be run?
ANOVA
if all the p values of the SW test are greater than alpha, but some or all p values of the Levene’s test are less than or equal to alpha, what test should be run?
Welch’s ANOVA
if any of the p values of the SW are less than or equal to alpha, what test should be run?
Kruskal Wallis test
what test is an expansion of the paired samples t test?
repeated measures ANOVA
what does the repeated measures ANOVA test do?
compares more than 2 repeated measurements on the same subject
what is the most common application for the repeated measures ANOVA?
pre, post, and follow up measurements of a subject
t/f: the repeated measures ANOVA uses the F-distribution
true
what is required of the F-distribution?
2 dfs
tests statistic F
what are the assumptions of the repeated measures ANOVA?
independence bw samples (not w/in)
continuous data
sphericity
normality
what is sphericity?
assumption that the difference bw time points have the same variance and correlated in approximately “the same way”
what is the test of sphericity?
Mauchly’s test of sphericity (used instead of test of variance of homogeneity)
how do we test normality most times with the repeated measures ANOVA?
using the graphical method (QQ plot)
why is the graphical method for normality testing used in repeated measures ANOVA?
bc you can’t find the difference for multiple groups
what are the hypotheses in repeated measures ANOVA?
H0: mut1=mut2=…=muti
Ha: some multi are different
the hypotheses for repeated measures ANOVA uses w/in or bw subjects hypotheses?
w/in subjects hypotheses
in the hypotheses for repeated measures ANOVA, what do mut1, mut2, mut3… mean?
mut1=population mean at time 1
mut2=population mean at time 2
mut3=population mean at time 3
etc
with repeated measures ANOVA, if the p>alpha, do we reject or fail to reject H0?
fail to reject H0 (accept H0)
with repeated measures ANOVA, if the p<alpha, do we reject or fail to reject H0?
reject Ho (accept Ha)
we have 18 subjects receiving a x ray fro chronic non-specfici back pain. the SF-36 (QOL measure) was measured for each subject b4 treatment, then at 1, 3, 6 months into treatment.
is there a difference in survey scores among the 4 points?
what is the hypothesis?
H0: mubaseline=mu1month= mu3month=mu6month
Ha: some multi are different
what are the 2 ways to assess normality in repeated sample ANOVA?
normality tresting on residuals: SST=SSR+SSE
graphical interpretation (QQ plot)
what is the QQ plot?
plots theoretical percentiles (quantiles) on the x axis vs sample percentiles on the y axis
if the plot and diagonal line of a QQ plot line up exactly, what does this mean?
the data is normal
what is the test of sphericity?
Mauchly’s test
if we assume the sphericity based on the Mauchly test, what data do we use in SPSS?
the sphericity assumed data
if we cannot assume sphericity based on the Mauchly test, what data do we use in SPSS?
the Greenhouse-Geiser data
what are the hypothesis of the Mauchly test of sphericity?
H0: sphericity
Ha: nonsphericity
if the p value of the mauchly test is greater than alpha, we do we conclude?
sphericity
if the p value of the mauchly test is less than alpha, what do we assume?
nonsphericity
how do we report statistics in the conclusion for repeated measures ANOVA?
(F(df1, df2)=f value, p=p value)
if the p value of the single factor repeated measures ANOVA is greater than alpha, do we have to run post hoc tests?
nope!
is the p value of the single factor repeated measures ANOVA is less than alpha, do we have to run post hoc tests?
yes!
what is the post hoc test for repeated measures ANOVA?
?????
t/f: when we run the post hoc for repeated measures ANOVA we have to use the 2 sided p values and compare them to the bonferroni corrected alpha
true
in post hoc testing, if a p value is less than the corrected alpha, is this significant?
yes!
in post hoc testing, if a p value is greater than the corrected alpha, is this significant?
no
what is the nonparametric alternative to (one-way) repeated measures ANOVA?
Friedman test
the Friedman test is an extension of the _____ test for paired samples
WSR
are we concerned about the sphericity of data with the Friedman test?
nope
when is the Friedman test indicated?
when data is n<30
when data is not normal (QQ plot)
when data is not categorical
what are the procedures of the Friedman test?
1) rank the values in each matched set (each subject) from low to high. (each subject is ranked separately)
2) add up the ranks for each of the groups/measurements (column)
3) calculate the mean ranks
4) calculate the test statistic Chi square
5) calculate the p value
does the chi square distribution use one or two DF?
just one
what are the hypothesis of the Friedman test?
H0: m1=m2=…=mk
Ha: some medians differ
are the hypothesis for the Friedman test one or two sided?
two sided
Researchers wish to know if the expired nitrogen during running differs depending on dietary conditions. A sample of 9 subjects received four dietary conditions (fasting, 23% protein, 32% protein and 67% protein) in a randomly chosen order. The outcome of interest was expired nitrogen (in liters) during running.
p=0.002
DF=3
chi square=15.133
what is the IV?
what is the DV?
what is H0?
what is Ha?
what test do we run?
do we run a post hoc?
if we run a post hoc, what is the bonferroni correction?
what is the post hoc test if needed?
IV: dietary conditions (% proteins in diet)
DV: expired nitrogen
H0: m1=m2=m3=m4
Ha: some mi are dif
run the Friedman test and find the observation ranks w/in each subject
conclusion: some mi are dif
run post hoc comparisons
Bonferroni correction: 0.05/6=0.00833
run the WSR test
if data is normally distributed based on the QQ plot, what test do we choose to run?
the repeated measures ANOVA
when we run the repeated measures ANOVA what test do we have to run to confirm sphericity?
Mauchly’s test
if Mauchly’s test shows that sphericity is violated, where do we get our F and p value from?
the Greenhouse-Geiser data
if Mauchly’s test shows that sphericity is confirmed, where do we get our F and p value from?
the sphericity assumed data
if the data is not normal in the QQ plot, what test should we run?
the Friedman’s test
if we have a sample data set with more than 2 independent samples, what parametric test do we run?
one way ANOVA
if we have a sample data set with more than 2 independent samples, what nonparametric test do we run?
Kruskal Wallis
if we have a sample data set with more than 2 related samples, what parametric test do we run?
repeated measures ANOVA
if we have a sample data set with more than 2 related samples, what nonparametric test do we run?
Friedman
