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What is the main question we want to answer with repeated measures ANOVA?
Are there differences in the mean scores of the dependent variable across groups/ conditions?
What is another name for repeated measures?
within-subject designs
What is a within-subjects design?
subjects are crossed with at least one experimental factor
a.k.a before and after treatment design (2 conditions)
each participant experiences each level of a factor/ each condition
what is a single-factor within-subject design
a one-way repeated measures design
n subjects are measured on the DV under k conditions (or levels) of a single IV or factor
ex. 8 children whose cortisol levels were measured 4 times in each child (pre, after 20 mins, after 40 mins, after 60 mins)
What are some possible research questions?
1. are there differences in the mean scores of the DV across groups/conditions?
within-subject effect of the IV (each subject is measured at each time point)
can be used to examine treatment effect of the IV (H0: μ1= μ2…=μk)
2. are there differences across subjects?
the variability of subjects (between- subject effect)
used to test the effect of the subjects (variance between subjects→ H0: Vs=0→ ex. all infants have the same level of cortisol)
One-way repeated measures ANOVA interests:
usually we are NOT interested in the effect of ‘subjects’ or subject-level variability (difference from between subjects)
if an effect exists, it would simply tell us that the subjects differ, but this has nothing to do with our treatment (IV) so it is irrelevant
we are interested in whether the IV has an effect on the subjects, regardless of whether differences existed naturally among subjects (factor A)
what is SS(T)?
Total sum of squares
what is SS(A)?
Variation between group means (IV)
What is SS(S)?
Variation between row means (=subject means)
what is SS (AxS)?
Variation between cell means (values within the cells)
Assumptions in one-way repeated measures ANOVA:
Normality: the distribution of observations on the dependent variable is normal within each level of the factor
Homogeneity of variance: the population variance observations is equal at each level of the factor (diagonal terms in a table are equal)
Homogeneity of covariance: the population covariance between any pair of repeated measurements is equal (homogeneous covariance)→ all off-diagonal terms are equal to eachother
ex. difference in variance between levels are the same →covariance (A,B)= covariance (A,C)= cov (B,C)
**Blue= compound symmetry = when both assumptions are met
Compound symmetry can be replaced with which assumption?
can be replaced by the assumption of sphericity: differences between levels of the factor, have the same variance of the differences for each level
In the assumption of sphericity, we assume that the relationship between pairs of experimental conditions is similar
Assumption of sphericity is necessary for validity of the F test in repeated measures ANOVA
What happens when sphericity is violated?
Tests: Mauchly’s W (1940)
H0: variances of differences between conditions are equal. (sphericity is met)
If p<0.05 (reject null=significant effect), the assumption of sphericity (and Compound symmetry) is violated
available in JASP
When compound symmetry is violated, the F-tests in one-way repeated measures ANOVA tend to be inflated, leading to more false rejections of H0
these violations require adjustments to the F test
Approaches to deal with violation of sphericity:
use a conservative critical value based on the possible violation of sphericity (conservative Ftest)
inflation of Ftest that occurs when sphericity is violated can be adjusted by evaluating the observed F value against a greater critical value, obtained by reducing the degrees of freedom.
DF (A)= Ɛ(k-1)→for factor in design→ with adjustment
DF(AxS)= Ɛ(k-1)(n-1) → for residual in design→ no adjustments=sphericity met then these are our DF for fcrit and f-observed
What are the statistical notations for when sphericity is met or violated?
sphericity met: Ɛ=1 (i.e., no correction needed)
sphericity violated: Ɛ<1 but can be anywhere from 0-1
when multiplied by epsilon, reduces DF (A) and DF (AxS), and gives a larger critical value for F-critical
How do we decide the value of Ɛ?
Ɛ > or equal to 1/ K-1 (lower bound=strictest correction or most extreme)
JASP provides two estimates of Ɛ
Greenhouse-Geisser (G-G) & Huynh-Feldt (H-F) estimates (correction epsilons)
G-G is smaller or more conservative → lower power to detect an effect→ corrects df more (stricter)
H-Y is closer to 1, higher power than G-G
both G-G and H-Y have higher power than using most extreme correction
what is compound symmetry?
the joint assumption of homogeneity of variance and homogeneity of covariance
what does Ɛ measure?
Ɛ measures the extent to which sphericity was violated (not computed manually)
what is the major difference between within-subjects and between-subjects ANOVA?
within-subjects is not concerned with SSs or the effects of subject variability
within-subjects is concerned with factor A
within subjects only has 1 f-ratio
what is the null hypothesis for the model effect of IV?
H0: μ1=μ2=…μk (depending on number of levels) all means of all the levels of factors are the same
enters computation of variation due to the model (SSA)
can also calculate effect of subjects (between participant effect) and treat each as a different level in an experimental design→ SSs but not relevant
what are the steps of the APA summary in repeated measures ANOVA?
summary (levels of factor, type of ANOVA)
Was sphericity met or not using Mauchly’s test? → state whether met and the exact p-value
state which measure of correction was used if sphericity not met→ H-F or G-G
The results (significant or not), state IV and DV, appropriate F statistic depending on the test used, P-value used, and partial w²
always mention what type of follow-up is needed if there are significant effects
what does SS(T) partition into in one-way reapeated measures?
SSs and SSw but we are only interested in SSw
what does SSw partition into?
SS(A) and SS(AxS)
what does SSA partition into?
MSA to then calculate the F ration (Fa)
What does MS(AxS) partition into?
FS and FA but we are only interested in the F-ratio for factor A
What does a Two-way ANOVA mean
comparing more than 2 means with two factors
each factor has two or more levels
multiple null-hypotheses concerning
main effect of Factor A
main effect of Factor B
interaction between Factor A and B
two way factorial experiments are…
experiments with TWO independent variables or factors
fully crossed: contain all possible combinations of the levels of factors (ex. 3×3, 2×2)
experiments that contain info about: two main effects + interaction effect
what is assumed about two-way factorial experiments
subjects serve in only one of the treatment conditions (independent sample or between-subjects designs)
if samples are equal in each condition it is called a balanced design
what are main effects?
the effect of one factor when the other factor is ignored (ex. main effect of factor A is a comparison to factor B)
the differences among marginal means for a factor(A and B)
what is an interaction effect?
The extent to which the effect of one factor depends on the level of the other factor:
present when the effects of one factor on the DV change at different levels of the other factor
indicates that the main effects of the factors do not fully describe the outcome of the factorial experiment
sometimes called crossover effect
considers pattern results for all cell means
What is the purpose of Two-way ANOVA
statistically examines the effects of:
two main factors of interest on the DV (main effects— row and column)
interaction between the different levels of these two factors (interaction effect)
what are the prior requirements/ assumptions for two-way ANOVA?
DV is normal within each group
variance of the population is equal for each group (homogeneity of variance)
independence of observations
Hypotheses for main effects:
main effect of factor A:
HoA: μA1= μA2…= μAa (equal row marginal means)
H1A: Not all μAg are the same (not all marginal means of factor A are equal to each other)
main effect of Factor B:
HoB: μB1= μB2…= μBb (equal column marginal means)
H1B: Not all μBj are the same (not all marginal means of factor B are equal to each other)
hypothesis for interaction effect:
H0, AXB: all μAgBj are the same OR the interaction between Factor A and Factor B is equal to zero
H1, AXB: not all μAgBj are the same OR the interaction between Factor A and Factor B is NOT zero
Basic concepts of two-way ANOVA
divides the variance observed into different parts resulting from different sources
assesses the relative magnitude of the different parts of the variance
examines whether a particular part of the variance is greater than expectation under the null hypothesis
What does SSM partition into in two-way ANOVA?
SSA: variation between means for Factor A
SSB: variation between means for Factor B
SSAxB: variation between cell means (interaction)
How to evaluate the F-ratio?
same way as usual
if the f-observed is greater than f-critical, we reject the null hypothesis
or look at p-value—If greater than or equal to 0.05, reject the null
calculating effect sizes
Use omega squared for each main effect and the interaction effect
what is the appropriate follow-up when we find a significant effect for the interaction (AxB)?
simple main effects because we can’t say anything about the main effects without knowing which factors we’re looking at
what is the follow-up when we find significant results for our main effects?
post-hoc tests
what does SS(T) partition into in two-way ANOVA?
SS(M) and SS(R)
How many effects are there in a three-way ANOVA?
7 effects:
3 main effects (A,B, C)
3 simple (two-way) interactions
1 three-way interaction (ABC)
What is the follow up when we have a significant two-way interaction
simple main effects
what is the follow up when we have a significant three-way interaction
Two-way ANOVAs→ pick one of the factors to do the two-way ANOVA and the remaining two are the factors for that ANOVA
what is the order in which we report significant effects in APA summary?
start with three-way if significant, then two-way and main effects last
what is the appropriate follow-up when factor B is significant
post hoc tests
what is the appropriate follow-up when we have a significant effect of factor AxB?
simple main effects
what is the appropriate follow-up when we have a significant three-way interaction?
two way ANOVA for 2 of the 3 factors at the level of the third factor
the two way interactions vary for each level of the third factor (ex. A and B vary based on level of C)
definitional formulas
represents the meaning of the parameter
ex. population variance states that it is the mean squared difference between a score and the mean of all scores
computational formulas
equations used to calculate values for a statistical concept
the definition of the construct may not be immediately obvious from the computational formula but it is algebraically equivalent (produces the same numerical value) to the definitional (conceptual) formula
generally used to speed up calculations
notation for computational formulas (one-way)
xij: the raw scores on the DV
i: individual and j: group
Treatment sums: A and Aj
Grand Sum: T
what is statistics good for?
answering research questions
what is a sample?
A subset of a population. Usually research focuses on populations
N= sample size
Difference between descriptive and inferential statistics
descriptive: summarize/describe properties of the sample (or the population)—>how are data distributed: central tendency, variability, shape of dist.
inferential (this class): draw conclusions regarding properties of the population, but based on sample data (i.e. comparing sample mean to population mean)
What 3 things do statistical analyses depend on?
study design/research
types of variables (levels of measurement, distribution)
whether assumptions of the analyses are met
Independent vs dependent variables
IV: predictor/covariate, factors in an experimental design
DV: outcome/response, predicted variables
correlational vs experimental research
correlational:
IV is measured by the researcher
no manipulation/ naturalistic
good for ecological validity
not good for inferring causality (i.e. confounding variable)
experimental:
IV is manipulated by the researcher
good for inferring causality
manipulating IV in lab settings may feel detached from real world
may have similar statistical methods to analyze data
not good for external validity
what is the first type of research?
between-subjects design: each participant is only in one experimental condition (i.e. control or treatment)
Groups should be equal on any confounding variables if random assignment.
allows generalization between population and sample
allows us to infer causality of the treatment
what is the second type of research?
within-subjects design: each participant does more than one experimental condition (i.e. control and treatment)
DV measured multiple times
limitation: vulnerable to practice effects (# of times repeated measure has been done) and fatigue/bordom effects and alternative explanations for differences between conditions
counterbalancing to rule out these alternative explanations of repeated measures designs
what are the 4 levels of measurments?
nominal: classifies objects (dichotomous=binary)
ordinal: categories with ranking
ratio: interval with true absolute 0—0 means lack of attribute
interval: rating data of equal distances, 0 means something
Are IV and DV continuous or categorical variables?
IV: categorical
DV: continuous (usually normally distributed)
Some key points about ANOVA:
Most used statistical method by experimental psychologists
categorical IVs (2+ categories)
can have more than 1 IV
a single continuous DV
usually assumed normally distributed
what are the measures of central tendency?
mean: most common
vulnerable to extreme values(outliers)
median: less vulnerable to extreme values (used when they are present)
mode:
unaffected by extreme values
used for numerical or categorical data (but mainly categorical/ nominal data)
may be none
may be several
what are the measures of variation?
range: simplest measure of dispersion
variance: SS/n-1 (n-1: less bias and closer to pop)
standard deviation: most common for descriptive stats
shows variation about the mean
same units as original data
square root of variance
the larger the standard deviation, the more variability
what are some ways to interpret variance and standard deviation better?
SD converted to z-score= better
if distribution is normal, SD can tell us about how many scores are above/ below a particular score
what is kurtosis?
the peak of the distribution
skewness:
left (negative) skew: mean < median
symmetric: mean=median
right (positive) skew: median > mean
what are the characteristics of a normal distribution?
DV expected to be continuous and normally distributed
mean=median=mode
population mean and standard deviation are sufficient to describe a normal distribution
contain about 68% of the values in pop
mean ± 2 SD = 95% of values in population or sample
mean ± 3 SD= 99.7% of values in population or sample
Hypotheses are assumptions about ___
population parameters NOT sample statistics (ex. studying 10 hours leads to better results than studying only 5 hours)
what are the steps for hypothesis testing (NHST)?
set up a hypothesis (H0 and H1)
choose alpha level, aka the cutoff value/critical value (0.05 unless told otherwise)
examine data and decide which statistical test to use (z,t,f)→ know when these are used
make a decision whether to reject or not reject the null hypothesis (i.e. whether the result is significant enough)
what does the p-value represent?
the probability of finding an effect in the observed statistic under the assumption that the null hypothesis is true
if p is less than or equal to a, reject the null
a statistically significant effect does not mean that…
we have a precise estimate of the effect (errors, effect of pop maybe smaller or larger than estimate, standard error of estimate)
the effect is important or meaningful (depending on the scale—the cutoff, it helps us determine whether there is a significant effect given our dataset)
what does a confidence interval give us information about?
the precision of our estimates
a CI should contain the population parameter but this doesn’t mean it always will
How does sample size influence the precisions of our estimates?
larger sample= more precise
smaller sample= less precise
CI beocmes more narrow =more precise
What happens to the CI as alpha decreases?
the CI becomes larger or wider (less precise)
What are some commonly used effect size measure in NHST?
pearson’s correlation r or r squared
cohen’s d
omega or omega squared
eta squared
what are considered small, medium and large effect sizes for pearson and cohen's d?
small: r=0.10, r² = 0.01, d= 0.2
medium: r=0.30, r²=0.09, d= 0.5
large: r= 0.50, r²= 0.25, d= 0.8
what are the two types of errors in hypothesis testing and define them
type 1 error (a): rejecting null hyp when it is true (false positive)
type 2 error (beta): retaining null hyp when it is false (false negative)
what is power (1-beta)
The probability of correctly rejecting a false H0
what is the relationship between alpha and beta?
Higher values of alpha (type 1 error) means lower values of beta (type 2 error), which means the power to reject the null is higher
the two are inversely related
what information do you need to compute a z-test
know population standard deviation or sample, sample size, and population mean or mean differences (x1/x2)
what is the purpose of a single mean z-test?
test whether the population mean is equal to some hypothesized value based on the sample mean that we have
what are the assumptions of a single mean z-test?
normal distribution
sample is simple and random (independence of observations)
the population standard deviation must be known
z score for sample score formula
z=x-μ/σ
what is the formula for z-statistic for sample mean?
z= x̄- μ0/σx̄
σx̄ : sample standard deviation (i.e. standard error)
σx̄= σ/√N
x̄: sample mean
μ0= population mean
what are the limitations of the z-test?
knowing the true population standard deviation is unrealistic (unless the entire population is known)
this is why t-tests are alternatives to z-tests
what is the purpose of a t-test?
to test whether the population mean is equal to some hypothesized value based on the sample mean
** we have no information about the population standard deviation
what are the assumptions/ requirements of t-tests?
the variable is normally distributed
the sample is simple and random (independence of observations—in no way influenced by measurements of other subjects)
what is the formula t-test statistic?
t= x̄- μ0/sx̄
μ0: hypothesized population mean (i.e. 0)
x̄: sample mean
sx̄= s/sqrt N (standard error)
who was the t distribution discovered by?
William S. Gosset in 1908 (a.k.a “student’s t distribution)
How does the t distribution vary in shape?
the shape varies according to the degrees of freedom (N-1)
+ sample size= +df
distributions are quite close to the normal distribution for df>30
as N +, t approaches z
Purpose of t-test with 2 means
To test whether 2 unknown population means (µ1 and µ2) are different from each other
H0: µ1 = µ2
H1: µ1 ≠ µ2
the two samples may be independent or correlated
What are independent samples/ between subject designs?
Each participant goes through 1 of the conditions in the experiment
What is a correlated sample/ dependent/within-subjects design/ repeated measures?
Each participant in the sample goes through all of the conditions in the experiment (not the same as independent samples)
what are the requirements for independent samples t-tests?
normally distributed in both populations
the standard deviations of the populations are the same (i.e. homogeneity of variance)
each subject is independent (simple, random sample
formula for independent samples t-test
t=x̄1-x̄2/ Sx̄1-x̄2=Dbar/SDbar
in other words, you calculate the mean difference of both group and subtract it by 0(null), then you divide it by the standard error of the mean difference
SDbar= pooled variance
Dbar= x̄1-x̄2
How do you calculate the SDbar or Sx̄1-x̄2 in the independent samples t-test
calculate the SS for each group
use formula: S²P=SS1+SS2/df1 + df2
find S.E: sqrt (S²P/n1 + S²P/n2)
calculate t-statistic
find the effect size using cohen’s d
**Df=N1-1 + N2-1
what is the formula for cohen’s d?
d=x̄1-x̄2/s
the mean difference divided by the the pooled standard deviation
s= pooled or estimated standard deviation (i.e. sqrt of s²)
**cohen’s d is most commonly used in t-tests
How do we calculate r using t-observed?
r=sqrt(t²/t²+df)
what is r²?
The proportion of variance in the Dependent variable that is explained by the independent variable
Note 
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