Advanced Stats: Final Exam
What is Stats?
"...a set of procedures and rules...for reducing large masses of data to manageable proportions and for allowing us to draw conclusions from those data" - objectively and quantifiably
We want to look at the ratio of differences _______________ from different conditions ______________ to variances from individual differences ___________________
(variances) (between groups) (within groups) If the ratio is large there is a significant impact from the condition (i.e. treatment effects), that is over and above the individual differences (noise)
Basic requirements of an ANCOVA
1 IV (nominal or ordinal) - discrete At least 2 levels 1 DV (interval or ratio) - continuous 1 CV (interval or ratio) - continuous
•Two IVs: Gender (male and female) and Age (young, medium, and old) •DV: performance
2x3 factorial anova
SD
A standardized measure of distance from the mean Average amount of variability in a set of scores, or Average distance from the mean Larger the SD, the larger the average distance each data point is from the mean of the distribution
Regression
A way of predicting the value of one variable from another. The model used is a linear one.
1) Now the above psychologists want to see if the treatment efficacy is affected by pre-treatment anxiety levels. They also give each participant the BAI (Beck Anxiety Inventory) and use these scores to control for anxiety levels. How should they analyze their data?
ANCOVA
1. Now we are going to control for social support as we know that that can affect depression. Run an ______________using the 4 groups on depr_diff with social as a covariate.
ANCOVA
ANCOVA - analysis of covariance
ANOVA with 1 (or more factors) and one DV and 1+ covariate (CV)
After the study?
Actual power calculation
Normal Curve
Almost 100% of the scores will fit between -3 and +3 SD from the mean Tails are asymptotic: Total area under the normal curve is set equal to 1.0
Per Comparison (PC) Error rate
Alpha (α) for each test
What is Alpha, and p?
Alpha = α = threshold for "significance" the probability of achieving the results That is your p value
Effect Size
An effect size is a standardized measure of the size of an effect: An effect size is a standardized measure of the size of an effect: Cohen's d Pearson's r Glass' Δ Hedges' g Odds Ratio/Risk rates
Ratio Scale
An interval scale with a true zero Identifies the direction and magnitude of differences and allows ratio comparisons of measurements
Interval Scale
An ordered series of equal-sized categories Identifies the direction and magnitude of a difference Zero is arbitrary on an interval scale.
Ordinal Scale
An ordered set of categories Tells you the direction of difference between two individuals labels that imply rank, doesn't say how much more one is than the other i.e. place in a race, 1st > 2nd > 3rd
All conditions in a repeated measures, one way ANOVA ___________
CROSS •ANOVA with 1 factor (one IV with 1+ levels) and one DV •And the levels "cross"
1) Imagine we found that drinking caffeine before going to bed significantly moderates the relationship between worry and sleep disturbances. What does this tell us?:
Caffeine effects the relationship between worries and sleep disturbance. Worrying may be an obstacle to sleeping, but becomes worse when caffeine is consumed. Therefore, caffeine is a moderator because if you had caffeine or not impacts the relationship between worry and sleep disturbance.
1) A social researcher wants to examine the effect of stereotype priming on evaluations of applications for employment. The researcher randomizes a group of people to two different groups: one group is shown a video clip that makes reference to a stereotype in workers, the other group is shown a neutral video about workplace hiring. The researcher then has both groups evaluate applications of employment and categorize them as "hire" or "don't hire". How would the researcher determine if the video had an effect?
Categorical DV, so it would be Chi-Square
Post Hoc Tests
Compare each mean against all others while controlling for Type 1 error These tests use a stricter criterion to accept an effect as significant Still a chance of Type 1 error, but it is reduced Hence, control the familywise error rate
Coeffiecient of Determination
Computed by squaring r varies between 0 and 1 (no negative values). It is % of variance in one variable (dv) that is accounted for by the variance in the other variable (iv) Proportion or % of variance shared by the 2 variables
Discrete/Categorical Variables
Consist of indivisible categories Naturally occurring groups or categories Ex: class size Ex: number of men and women, percentage of people with a given hair color
Dependent T-tests
Dependent t (sometimes call paired t-test, pre-post) Compares two means based on related data 1 group, 2 means
dependent t test
Dependent t (sometimes called "paired" or "matched t-test", or "pre-post") Compares two means based on related data. 2 means, 1 group Usually the same people in both groups or people/scores are related.
Assumptions of t tests
Dependent variables are interval or ratio. population is normally distributed. Samples are randomly selected. The groups have equal variance (Homogeneity of variance). The t-statistic is robust (it is reasonably reliable even if assumptions are not fully met)
Descriptive Stats
Descriptive statistics are methods for organizing and summarizing data A descriptive value for a population is called a parameter and a descriptive value for a sample is called a statistic
Accepting the HA =
Difference among the means Simply establishes that differences exist!
PC < _____< PE
FW
Independent Errors:
For any pair of observations, the error terms should be uncorrelated.
Homoscedasticity:
For each value of the predictors the variance of the error term should be constant.
"I noted earlier that the ____________ correction is probably too strict and that the ____________ correction is probably not strict enough"
Greenhouse-Geisser Huynh-Feldt
ANOVA with 1 factor (one IV with 1+ levels) and one DV
IV also called factor, where the number of groups is referred to as number of levels
Imperfect Relationship of Y and X
Imperfect relationship between Y and X: Y= b0 + b1X + e e = error of estimation
Familywise (FW) Error Rate
In a "family" of comparisons between means, we can estimate the probability that we have at least 1 Type I error in our family of comparisons FW = 1- (1-PC)c
Independent T-tests
Independent samples t (or two samples t test) Compares two means based on independent data 2 groups, 2 means no relation between groups e.g., people randomly assigned to each group
Inferential Stats
Inferential statistics makes inferences and predictions about a population based on a sample of data taken from the population in question
Continuous/Dimensional Variables
Infinitely divisible into whatever units Ex: time or weight Scores that provide information about the magnitude of differences between participants Ex: level of anxiety symptoms, scores range from 0-20
For mediation to occur
Iv must predict DV IV must predict mediator Mediation must predict DV The relationship between IV and DV should be small when mediator is included in the model than when it isn't
Sample to Pop
Mean and SD are obtained from a sample, but are used to estimate the mean and SD of the population (very common in psychology).
Cohen's d
Measure of how different two groups are from one another, or the magnitude of effect.
The Semi-Partial Correlation
Measures the relationship between a predictor and the outcome, controlling for the relationship between that predictor and any others already in the model. It measures the unique contribution of a predictor to explaining the variance of the outcome.
1) Some 3rd wave psychologists want to determine which of ACT, CBT, or DBT is superior in treating depression. They randomly assign participants to each treatment group and give them the BDI before treatment and after treatment. What is the best way to analyze this data? (3 treatment conditions, 1 factor, 3 levels. Repeated measures).
Mixed Model Designs
1) A university wants to predict which students might drop out some time after their first year. They use high school SAT scores and 1st year college GPA of their past students to build a model that predicts who drops out after their first year (drop-out or not drop-out). They use this model to predict which freshman are vulnerable to dropping out and offer them more services and support for staying in school. How do they build the model?:
Multiple Linear Regression
•A record company boss was interested in predicting album sales from advertising. •Data •200 different album releases •Outcome variable: •Sales (CDs and Downloads) in the week after release •Predictor variables •The amount (in £s) spent promoting the album before release (see last lecture) •Number of plays on the radio (new variable)
Multiple Regression
Multiple Regressions
Multiple Regression is a model to predict values of an outcome from several predictors. It is a hypothetical model of the relationship between several variables.
What tests can we use to see if mean differences are significant?
Multiple t-tests, but Inflates the Type I error rate (more on this next) Orthogonal Contrasts/Comparisons Hypothesis driven Planned a priori Post Hoc Tests Not Planned (no hypothesis) Compare all pairs of means
Accepting the H0 =
No difference among the means
ANCOVA Assumptions
Normality Homogeneity of variance All measurements are independent Relationship between DV and covariate is linear (and absence of multicollinearity) The relationship between the DV and covariate is the same for all groups (all levels of the IV)
independent samples t test
Note that we aren't asking or saying which is larger or smaller - just that they are different
Simple ANOVA
One-way ANOVA Testing for difference between scores of 2 or more different groups F test is the test statistic
Method of Least Squares
Ordinary least square (OLS) or line of best fit OLS is the technique used to estimate a line that will minimize the error (the difference between the predicted and the actual values of Y)
Love of Puppies ex
Outcome (or DV) participant's happiness Predictor (or IV) Dose of puppy therapy (control, 15-mins & 30-mins) Covariate love of puppies
1. Let's first see if our peeps got better from pre to post. a. Run a _________________ with dpr_pre and dpr_post, assuming all our participants are in one group.
Paired-samples T-test
Partial and Semi-Partial Correlations
Partial correlation: Measures the relationship between two variables, controlling for the effect that a third variable has on them both. Semi-partial correlation: Measures the relationship between two variables controlling for the effect that a third variable has on only one of the others.
Perfect Relationship of Y and X
Perfect relationship between Y and X: Y= b0 + b1X X causes all change in Y
Skewed Histograms
Positive skew (scores bunched at low values with the tail pointing to high values). Also called skewed right. Negative skew (scores bunched at high values with the tail pointing to low values). Also called skewed left.
Before the study?
Power analyses
Non-Zero Variance:
Predictors must not have zero variance.
Pros and Cons of Hierarchical Regressions
Pro: It is the best method for theory testing. Can see the unique predictive influence of a new variable on the outcome (because known predictors are held constant in the model, i.e. entered before). Con: Relies on the experimenter knowing what they are doing!
Advantages of ANCOVA?
Reduces Error Variance By explaining some of the unexplained variance, the error variance in the model can be reduced Greater Experimental Control: By controlling known extraneous variables, we gain greater insight into the effect of the predictor variable(s)
Properties of the Linear Regression Line
Represents the predicted values for Y for any and all values of X. Always goes through the point corresponding to the mean of both X and Y. It is the best fitting line in that it minimizes the sum of the squared deviations. Has a slope that can be positive or negative; null hypothesis is that the slope is zero.
Repeated measures ANOVA
Same group of people is tested under different conditions •Condition1, condition2, condition3, condition4 •Everybody is subjected to ALL the different drugs/treatments/therapies/conditions •Repeated measures or within subjects •No between subjects here
Sampling Distribution of the Mean
Sampling distribution is a distribution of a statistic (the means, not raw data) over all possible samples
Bonferroni's test (Dunns)
Set your alpha to be .05 divided by the total number of tests made Extremely conservative Reduces power as the number of comparisons being made increase
degrees of freedom
So if we know the total, all but one score is free to vary. That last score is predetermined by the other scores.
Variance
Standard deviation squared SD vs. Variance SD is stated in the original units Variance is harder to interpret because based on squared deviation scores
Bias
Systematic differences between the characteristics of a sample and a population
For a casual mediation relationship, you have to show ________
Temporal precedence •Predictor occurs/measured before mediator •Mediator occurs/measures before outcome •Often looking for predictor related changes in mediator, and those changes in mediator related changes in outcome
Per Experiment (PE) Error Rate
The amount of Type I errors we expect to make when the null hypothesis is true
R
The correlation between the observed values of the outcome, and the values predicted by the model.
One Way ANOVA Assumptions
The data are sampled randomly and independently of each other Homogeneity of variance variances of each sample are assumed equal Levine's test Normality
Sampling Error
The discrepancy between a sample statistic and its population parameter is called sampling error
What is an extraneous variable in an ANCOVA?
The extraneous variable is the covariate - it co-varies with the DV
Correct Interpretation of P value
The p value is the probability of getting a value of a test statistic as or more extreme than the value of the statistic computed from the collected data, under the assumption that the null hypothesis is true
R^2
The proportion of variance accounted for by the model.
Standard Error
The standard deviation of the sampling distribution is the standard error. For the mean, it indicates the average distance of the statistic from the parameter
Linearity:
The underlying relationship we model is linear.
If the difference between the samples is larger than what we would expect based on the standard error, then we can assume one of two:
There is no effect and sample means in our population fluctuate a lot and we have, by chance, collected two samples that are atypical of the population from which they came. The two samples come from different populations but are typical of their respective parent population. In this scenario, the difference between samples represents a genuine difference between the samples (and so the null hypothesis is incorrect).
The problem with means
They are sensitive to outliers
Why use ANCOVA?
To test for differences among group means when we know that an extraneous variable affects the outcome variable Used to control known extraneous variables that cannot be controlled for in other ways (random assignment or other experimental control)
Multicolinearity
Tolerance should be more than 0.2 (Menard, 1995) VIF should be less than 10 (Myers, 1990) Multicollinearity exists if predictors are highly correlated. This assumption can be checked with collinearity diagnostics.
Type 1 and Type 2 Errors
Type I error occurs when we believe that there is a genuine effect in our population, when in fact there isn't. Type II error occurs when we believe that there is no effect in the population when, in reality, there is.
Nominal Scale
Unordered set of categories identified only by name Labeling/classifying objects, i.e. categorical Not technically a scale of measurement since nothing is measured
Pearson's r
Used to describe the strength of the linear relationship between two quantitative variables
Stratified Sampling
Used to ensure that the proportional representation of groups in the sample is the same as in the population.
Variability
Variability reflects how scores differ from one another 3 typical measure of variability Range Standard deviation Variance
Assumptions about Regressions
Variable Type: Non-Zero Variance: Linearity: Independence: Predictors must not be highly correlated. independent Errors: Normally-distributed Errors
Sum of Squares
We get the Sum of Squared Errors (SS) Indicates the total dispersion, or total deviance of scores from the mean:
Covariance
We need to see whether as one variable increases, the other increases, decreases or stays the same (COVARIANCE). We look at how much each score deviates from the mean. If both variables deviate from the mean by the same amount, they are likely to be related
Z Score
What is a z score? A measure of an observation's distance from the mean - Standardized in units of SD
What means would we expect to obtain if there was no effect?
What means would we expect to obtain if there was no effect? i.e. if they came from same population i.e. if the null hypothesis were true Expect no difference
Calculating Sampling Error
When not a perfect fit..... (all of the time) A deviation is the difference between the mean and an actual data point Deviance = score - mean
ANCOVA's adjusted means
When using ANCOVA, the means for each group get adjusted by the CV-DV relationship If the covariate has a significant relationship with the DV, than any comparisons are made on the adjusted means
Variance can be separated into two major components
Within group variability or differences within a particular group i.e. individual differences, normal "noise" we would expect by chance/error Between group differences depending what one group experiences vs. another i.e. group "condition"
1) Now imagine we found a relationship between listening to soothing music before bed and sleep disturbances. If we hypothesize that listening to soothing music before bed reduces worrying and that might affect sleep disturbance, what would we call "worrying" in this case? How would you write out the relationship (in English)?
Worrying in this case would be classified as a mediator in this relationship between the soothing music (IV), and sleep the (DV). Listening to soothing music reduces worry and effects sleep. The amount of worry (mediator) reduces the relationship between soothing music and sleep. There was an indirect effect of soothing music on sleep through the level of worry.
Pearson's R
X = predictor/independent; Y = outcome/dependent Sign provides information about the direction of the relationship between X and Y Absolute value reflects the strength of the correlation Reflects the amount of variability that is shared between two variables and what they have in common
reducing bias
analyse w/ robust methods - non parametric tests - bootstrapping - methods based on medians + trims trim the data windsorizing transform the data
•There are several possible confounding variables - e.g. love of puppies. •We can replicate the RCT of puppy therapy but also measure love of puppies. •Outcome (or DV) •participant's happiness •Predictor (or IV) •Dose of puppy therapy (control, 15-mins & 30-mins) •Covariate •love of puppies
ancova
ANCOVA Pairwise comparisons are
based on the estimated marginal means
transform the data
by applying a mathematical function to scores
•Can animals be trained to line-dance with different rewards? •Participants: 200 cats •Training •The animal was trained using either food or affection as reward •Dance •The animal either learned to line-dance or it did not. •Outcome: •The number of animals (frequency) that could dance or not in each reward condition. •We can tabulate these frequencies in a contingency table.
chi squared
trim the data means
delete a certain amount of scores form the extremes
•Are invisible people mischievous? •12 Participants •Manipulation •Placed participants in an enclosed community riddled with hidden cameras. •For first week participants normal behavior was observed. •For the second week, participants were given an invisibility cloak. •Outcome •measured how many mischievous acts participants performed in week 1 and week 2.
dependent samples t-test (matched samples t-test)
MANOVA Assumptions
each cell (group) must have a minimum number of subjects being equal to, or greater than, the number of DVs. •Normality for both univariate normality AND multivariate normality •Multivariate assessed using the Mahalanobis distances. •Outliers •Check for univariate outliers •Also check for multivariate outliers •Linearity •Linear relationship between each pair of the DVs - best if of medium strength •Multicolinearity •Correlations between each set of pairwise DVs •If r > .8, then the assumption of multicollinearity is violated •Singularity •Each DV has to be independently separate from other DVs (not a combination of two) •Homogeneity of variance-covariance matrices and regression
Run a _________________ with meds_vs_therapy (medication versus talk therapy) and gender (male, female) as IVs and dpr_diff (change in depression scores pre-post) as the DV.
factorial ANOVA
•Beer-goggling •Attractiveness rating of face-types (pre-rated attractive and unattractive faces) at different levels of alcohol consumption •IVs: •Face-types •2 levels (attractive, unattractive) •Alcohol consumption •3 levels (Placebo, low-dose, high-dose) •DV •Attractiveness ratings
factorial anova
•If statistical power is _________, the probability of making a Type II error, goes down. •i.e. probability of concluding there is no effect when, in fact, there is one, goes down
high
What means would we expect to obtain if there was an effect?
i.e. if came from different population i.e. if the alternative hypothesis were true Expect there to be a difference Although it is possible for their means to differ by chance alone, we would expect large differences between sample means to occur very infrequently.
Now let's compare how the meds group did in comparison to the therapy group. Run an _______________ with med_vs_therapy as the grouping variable and depr_diff as the depedent variable.
independent sample t-test
•Are invisible people mischievous? •24 Participants •Manipulation •Placed participants in an enclosed community riddled with hidden cameras. •12 participants were given an invisibility cloak. •12 participants were not given an invisibility cloak. •Outcome •measured how many mischievous acts participants performed in a week.
independent samples t-test
Independent variable (IV)
is the categorical variable being used to define the groups i.e. the variable the experimenter manipulated Notation: k is the number of groups
Dependent variable (DV)
is the group scores (means) you are comparing Continuous variable
Multiple T Tests
k is number of groups or means If we did all the possible contrasts (k)(k -1)/2
On the other hand, a large treatment effect will produce a _________ value for the F-ratio.
large Thus, when the sample data produce a large enough F-ratio, we will reject the null hypothesis and conclude that there are significant differences between treatments/conditions
assessing homogeneity of variance
levene's tests: tests if variances in different groups are the same you want levene's test to be greater than .05 in order to meet the assumption
•A record company boss was interested in predicting album sales from advertising. •Data •200 different album releases •Dependent/Outcome/criterion variable: •Sales (CDs and Downloads) in the week after release •Independent/Predictor variable: •The amount (in £s) spent promoting the album before release.
linear regression
•Predictors of a treatment intervention. •Participants •113 adults with a medical problem •Outcome: •Cured (1) or not cured (0). (only 2 levels, thus binary) •Predictors: •Intervention: intervention or no treatment •Duration (of illness): the number of days before treatment that the patient had the problem
logistic regression
•Different types of therapy on 2 PTSD symptoms •Three Groups: •Eye-Movement Desenstizing Reprocessing Therapy (EMDR) •Cognitive Processing Therapy (CPT) •Prolonged Exposure (PE) •Two Outcome Variables (DVs): •Hypervigilance •Frequency of flashbacks
manova
•Efficacy of psychotherapy on OCD •Three Groups: •Cognitive Behaviour Therapy (CBT) •Behaviour Therapy (BT) •No Treatment (NT) •Two Outcome Variables (DVs): •Obsession-related Actions •Obsession-related Thoughts
manova
assumption of normality
means of sampling distribution and the data inside each sample are normal this matters in small samples, can be forgotten in larger samples
•initial relationship is between pornography consumption (the predictor) and infidelity (the outcome) •hypothesized that this relationship is mediated by commitment (the mediator) •This model suggests that the relationship between pornography consumption and infidelity isn't a direct effect but operates though a reduction in relationship commitment
mediation
• A number of studies have found that marital violence is positively related to child aggression. • According to the spill-over hypothesis, this association is mediated by negative parenting. • The negativity of the marital dyad spills over the parent-child dyad which then influences child behavior. Thus, negative parenting helps explain why marital violence is related to child behavior.
mediator
•Example •Between-subjects IV: Group/condition •2 Levels: Meds, talk therapy •Within-subjects measure: Time •Measures taken pre and post •DV: Depression scores
mixed model designs
•Example •Between-subjects IV: Group/condition •2 Levels: Meds, talk therapy •Within-subjects measure: Time •Measures taken at pre, post •Covariate: Social support level •DV: Depression scores
mixed model designs with a covariate
•Do violent video games make people antisocial? •Participants •442 youths •Outcome •Aggression •Callous unemotional traits (CaUnTs) •Number of hours spent playing video games per week
moderation
•For instance, imagine researchers are evaluating the effects of a new anti-depressant. The researchers vary the participants by those who are currently doing talk therapy and those who are not. They measure their depression levels after 6 weeks. •They find that for those not in talk therapy, there is a small association between the drug and depression levels, but for those also doing therapy, there is a huge association between the drug and depression levels
moderation
• A number of studies have found that family adversity (e.g., negative parenting) is positively related to child aggression. • However, studies (Criss et al., 2002; Lansford et al., 2003) indicated that positive peer relationships moderate the link between family adversity and child aggressive behavior. • Specifically: • Under HIGH positive peer relationship: FA ® AGG = ns • Under LOW positive peer relationship: FA ® AGG = sig • Thus, when is family adversity significantly related to child aggression? • Answer: when children have poor peer relationships. • Criss (2001)
moderator
•A puppy therapy RCT in which we randomized people into three groups: 1.A control group 2.15 minutes of puppy therapy 3.30 minutes of puppy contact •The outcome is happiness (0 = unhappy) to 10 (happy). •Predictions: 1.Any form of puppy therapy should be better than the control (i.e. higher happiness scores) 2.A dose-response hypothesis that as exposure time increases (from 15 to 30 minutes) happiness will increase too
one way anova
•Ex. Testing a new intervention to reduce anxiety •Measure participants' anxiety levels pre-intervention, post-intervention, and at 6-month follow-up.
one way repeated measures anova
•Longitudinal study of annual growth (height) of children over the first five years of life? •IV: •1 factor: Time •5 levels: year1, y2, y3, y4,y5 •DV: •Height measurement •
one way repeated measures anova
•Same group of people is tested under different conditions •Condition1, condition2, condition3, condition4 •Everybody is subjected to ALL the different drugs/treatments/therapies/conditions
one way repeated measures anova
•The effects of 0 beers, 2 beers, & 5 beers on breaking reaction time of the same participants, over the course of 3 days? •IV: •1 factor: dose •3 levels: None, 2 beers, 5 beers •DV: •Breaking reaction time
one way repeated measures anova
•The measurement of depression before therapy, after therapy, and 9-month followup? •IV: •1 factor: Time •3 levels: Before therapy, after therapy intervention, 9 month followup •DV: •Measure of depression
one way repeated measures anova
1. Now let's see if there are significant differences among our four groups: SSRI, SNRI, CBT, ACT. Run a _________________ with treatment_group as the IV and depr_diff as the DV.
one-way ANOVA
Decision Rule
p ≤ α statistically significant evidence We reject H0 and accept H1 p > α nonsignificant evidence We reject H1 and accept H0
Ivs and DVs are called what in regressions?
predictor (IV) and criterion (DV) variables
1. Run a ________________with 3 time points: dpr_pre (depression scores at the start of the study), dpr_post (depression scores at the end of the treatment), and dpr_6mon (depression scores at 6-month follow-up). You can name your factor "time". And the dependent variable is "depression".
repeated measures ANOVA
•4 bizzaro animal parts •Stick insect •Kangaroo Testicle •Fish Eyeball •Witchetty Grub •Each participant is asked to eat each of the animal parts, and time to retching is measured.
repeated measures one way anova
windsorizing
substitute outliers w/ the highers value that isn't an outlier
Standardized beta values:
tell us the same but expressed as standard deviations.
Beta values
the change in the outcome associated with a unit change in the predictor.
The Sum of Squares, Variance, and Standard Deviation represent ____________________
the same thing: The 'Fit' of the mean to the data The variability in the data How well the mean represents the observed data Error
If the "treatment effect" is zero (no differences between groups), the top and bottom of the F-ratio are measuring ______________
the same variance (error/chance/within group variance) In this case, you should expect an F-ratio near 1.00
homoscedasticity
the variance of y for each value of x is constant (correlations and predictions)
F =
treatment effect + chance/error F = ────────────────────── chance/error
homogeneity of variance
variability w/in two or more samples are about the same (F-tests, and T-tests)
What are the requirements to use a MANOVA
•"It is a bad idea to lump outcome measures together in a MANOVA unless you have a good theoretical or empirical basis for doing so" •Best when DVs are moderately correlated •.3 < r < .8. is a good range • •If r is greater than 0.8, then the results of the MANOVA can be biased (almost like DVs are no longer independent) •If r is < .3 then no advantage to MANOVA •In these situations: better to do separate ANOVAs
Mediation
•A mediating variable explains the relationship between the independent (predictor) and the dependent (criterion) variable. •It explains how or why there is a relationship between two variables. • A mediator can be a potential mechanism by which an independent variable can produce changes on a dependent variable.
Moderation
•A moderator is a variable that affects the strength of the relation between the predictor and criterion variable. •Moderators specify when a relation will hold. It can be qualitative (e.g., sex, race, class...) or quantitative (e.g., drug dosage or level of reward). •In other words, which variables influence treatment effects •for whom does the intervention work? •(e.g., matching) • •Moderating variables are typically an interaction term in statistical modelsRe
What is a regression?
•A way of predicting the value of one variable from another. •It is a hypothetical model of the relationship between two variables. •The model used is a linear one. •Therefore, we describe the relationship using the equation of a straight line.
Ancova
•ANCOVA - analysis of covariance •ANOVA with 1 (or more factors) and one DV •and 1+ covariate (CV) •1 IV (nominal or ordinal) - discrete •At least 2 levels •1 DV (interval or ratio) - continuous •1 CV (interval or ratio) - continuous
•Factorial ANOVA (or two-way, or 3-way, etc)
•ANOVA with 2 or more factors (2+ IV's, each with 1+ levels, and one DV)
•MANOVA
•ANOVA with MORE than 1 DV (1+ IVs and 2+ DV)
•MANCOVA
•ANOVA with MORE than 1 DV and 1+ covariates (1+ IVs and 2+ DV)
Why is power important?
•As researchers, we put a lot of effort into designing and conducting our research. This effort may be wasted if we do not have sufficient power in our studies to find the effect of interest. •Sample size too big; too much power wastes money and resources on extra subjects without improving statistical results •Sample size too small; having too little power to detect meaningful differences •Treatment discarded as not important when in fact it is useful... •Improving your research design Improving chances for funding
Mixed model designs with covariate
•Between groups and within groups design •AND a covariate •Example •Between-subjects IV: Group/condition •2 Levels: Meds, talk therapy •Within-subjects measure: Time •Measures taken at pre, post •Covariate: Social support level •DV: Depression scores
Mixed model designs
•Between groups and within groups design •Example •Between-subjects IV: Group/condition •2 Levels: Meds, talk therapy •Within-subjects measure: Time •Measures taken pre and post •DV: Depression scores
Reporting moderation: example
•Callous traits moderates the association between hours playing video games and aggressive behavior. •For fewer hours played, there is no difference in aggressive behavior between those with or without callous traits •However, for those playing for a lot of hours, there is a large difference between those with and without callous traits and aggressive behaviors, where those with callous traits have much higher level of callous behavior compared with those who don't.
Chi-square (χ2)
•Chi-square analysis is primarily used to deal with categorical data •Investigates whether distributions of categorical variables differ from one another •Nonparametric test •categorical variables do not have means or standard deviations •Uses frequency data (counts)
dependent t-test
•Dependent t (sometimes call paired t-test, pre-post) •Compares two means based on related data •1 group, 2 means •Dependent variables are interval or ratio.
Interactions
•Effects of one independent variable differ according to levels of another independent variable
factorial anova
•Factorial ANOVA (or two-way, or 3-way, etc) •ANOVA with 2 or more factors (2+ IV's, each with 1+ levels, and one DV)
Moderation: Example
•For instance, imagine researchers are evaluating the effects of a new anti-depressant. The researchers vary the participants by those who are currently doing talk therapy and those who are not. They measure their depression levels after 6 weeks. •They find that for those not in talk therapy, there is a small association between the drug and depression levels, but for those also doing therapy, there is a huge association between the drug and depression levels •Talk therapy moderates the association between drug effect and depression levels
•Complete/full mediation
•Full: •Complete mediation is present when the independent variable no longer influences the dependent variable after the mediator has been controlled.
Advantages of a Factorial ANOVA
•Increased power •with the same sample size and effect size, a factorial ANOVA is more likely to result in the rejection of Ho •And it follows, with equal effect size and probability of rejecting Ho if it is true (α), you can use fewer subjects (and time and money) • •AND........factorial ANOVA can detect interactions
independent samples t-test
•Independent samples t (or two samples t test) •Compares two means based on independent data •2 groups, 2 means •no relation between groups •e.g., people randomly assigned to each group •Dependent variables are interval or ratio.
•Power = 1 - β
•Inversely related to β or the probability of making a Type II error •if b =.20 then power =.80; we will accept a 20% chance of missing an association of a particular size b/w an exposure and an outcome if one really exists
Mediation: example
•Lambert et al. mediator model: •initial relationship is between pornography consumption (the predictor) and infidelity (the outcome) •hypothesized that this relationship is mediated by commitment (the mediator) •This model suggests that the relationship between pornography consumption and infidelity isn't a direct effect but operates though a reduction in relationship commitment
mancova
•MANCOVA •ANOVA with MORE than 1 DV and 1+ covariates (1+ IVs and 2+ DV)
manova
•MANOVA •ANOVA with MORE than 1 DV (1+ IVs and 2+ DV)
Testing Sphericity
•Mauchly's Test •We want non-significance p > .05 •Assumption of equality of variances is met •If p < .05 •then the data fails to meet the assumption of sphericity •Need to use one of the correction factors •Greenhouse-Geisser estimate (more conservative) •Huynh-Feldt estimate •Or a more conservative test •multivariate
What is a multiple regression?
•Multiple Regression is a model to predict values of an outcome from several predictors. •It is a hypothetical model of the relationship between several variables.
Main Effects
•One independent variable has a significant effect on the dependent variable, regardless of the levels of the other independent variable(s)
one way anova
•One-way ANOVA •Testing for difference between scores of 2 or more different groups •with 1 factor (one IV with 1+ levels) •and one DV •Independent variable (IV) •is the categorical variable being used to define the groups •Dependent variable (DV) •is the group scores (means) you are comparing •Continuous variable
Partial mediation
•Partial: •Partial mediation occurs when the independent variable's influence on the dependent variable is reduced after the mediator is controlled.
Logistic Regression example
•Predictors of a treatment intervention. •Participants •113 adults with a medical problem •Outcome: •Cured (1) or not cured (0). (only 2 levels, thus binary) •Predictors: •Intervention: intervention or no treatment •Duration (of illness): the number of days before treatment that the patient had the problem
Where does effect size come from?
•Previous research! •What effect size have those the come before you found? •Look at articles that have similar questions and measures, and the data are analyzed in the same way you want to • •You might need to convert effect sizes •You need to decide - use minimum possible effect size? An average? •This affects how many people you need to recruit
one way repeated measures anova
•Same group of people is tested at different times •Pre, post, follow-up (Time1, Time2, T3) •Everybody got the same drug/treatment/therapy/condition •Interval scale data in each group •ANOVA with 1 factor (one IV with 1+ levels) and one DV •And the levels "cross"
Tradeoffs with Sample Size
•Sample size is affected by effect size, a, b, power •If detected effect size is then sample size ¯ •If detected effect size is ¯ then sample size • •Power increases as N increases. •The more independent scores that are measured or collected, the more likely it is that the sample mean represents the true mean.
Interpreting Interactions
•Significant interaction term tells us they differ, but we don't know how they differ •The most simple and common method of interpreting interactions is to look at a graph•
Categorical Data
•Sometimes we have data consisting of the frequency of cases falling into unique categories •Examples: •Number of people voting for different politicians •Numbers of students who pass or fail their degree in different subject areas •Number of patients or waiting list controls who are 'free from diagnosis' (or not) following a treatment
Chi-square assumptions
•The chi-square test has two important assumptions: •Independence: •Each person, item or entity contributes to only one cell of the contingency table. •The expected frequencies should be greater than 5 in each cell of contingency table
What is power?
•The likelihood that a study will detect an effect when there is an effect there to be detected •i.e. probability that a study will reject a false null hypothesis
What would it look like if there was no interaction?
•The lines are parallel and do not cross. •Remember if the lines cross (on or off the graph) then there is an interaction.
Reporting mediation: example
•There was a significant indirect effect of pornography consumption on infidelity through relationship commitment. Or •It was found that relationship commitment partially mediated the relationship between pornography consumption and infidelity.
How do we follow up a MANOVA?
•This is the traditional/typical way to follow up a MANOVA -> with post hoc contrasts. •Field says that discriminant analysis is better •Reference section in MANOVA chapter
How to do see if there was an interaction?
•To interpret an interaction, we graph the DV on the Y-axis, place one IV on the X-axis, and define the lines by the other IV •Lines cross one another •Hallmark of an interaction •However, it is not necessary that the lines cross, only that the slopes differ from one another •i.e. they would cross if continued to infinity
Logistic Regression: When and Why
•To predict an outcome variable that is categorical from one or more categorical or continuous predictor variables • •Used because having a categorical outcome variable violates the assumption of linearity in normal regression • •Outcome •We predict the probability of the outcome occurring
Why would we use a MANOVA
•To test for differences between groups •When we have several outcome variables (DVs) •(as long as the DVs are logically and properly correlated) • •Better than Multiple ANOVA •Controls familywise error rate (Type I error) •Takes account of relationships between DVs
Chi-Square summary
•Two categorical variables •Pearson's chi-square test •Three or more categorical variables: •Loglinear model (examples in book) •For every variable we get a main effect and interactions Effect Sizes •The odds ratio is a useful measure of the size of effect for categorical data
Factorial ANOVA
•Why use a factorial ANOVA? Why not just use multiple one-way ANOVAs? •With 3 IVs, you'd need to run 3 one-way ANOVAs, which would inflate your family-wise α-level
•A MANOVA reduces the chances of making a Type I error by running fewer analyses
•instead of running three separate ANOVAs each with one DV, run only one MANOVA •Under certain conditions, a MANOVA may find differences that do not show up under separate ANOVAs
Data in each group of a repeated measures, one way anova
•interval scale •Normally distributed •Homogeneity of variance •Sphericity •Variance of the differences between conditions is the same Variance t1-t2 » Variance t1-t3 » Variance t2-t3 •Commonly violated (still okay to use with corrections....)
Factorial ANOVA Notation
•number of numbers = the number of factors (or IVs) •numbers themselves = the number of levels in each factor (IV)
Statistical Power is affected by?
•the size of the effect •the size of the sample •Bigger effects are easier to detect than smaller effects •Larger samples are usually needed if the effect we want to detect is small•Statistical power is affected by: •the size of the effect •the size of the sample •Bigger effects are easier to detect than smaller effects •Larger samples are usually needed if the effect we want to detect is small
