Advanced Statistics Exam 3
Four step Process for hypothesis testing with the repeated measures t statistic
1. state the hypothesis. Select the alpha level 2. Locate the critical region 3. calculate the t statistic 4. Make a decision
A research design with 2 factors
2x3 design 2 Factors are the groups: Therapy 1: Group 1 and Therapy II: Group II 3 levels or time variables: Scores for before therapy, time after therapy, and 6 months after therapy.
New Notation for the repeated measures ANOVA
P=represents the total of all the scores for each individual in the study. (Person Total)
M
sample mean from the data
ANOVA notation n
the number of scores in each treatment
The F distribution Table
*For ANOVA, we expect F near 1.00 if H0 is true. (an F ratio that is much larger than 1.00 is an indication that H0 is not true. *In the F distribution, we need to separate those values that are reasonably near 1.00 from the values that are significantly greater than 1.00. -These critical values are presented in an F distribution table. To use the table, you must know the df values for the F-Ratio (numerator and denominator) and you must know the alpha level for the hypothesis test. -It is customary for an F table to have the df values for the numerator of the F ratio printed across the top of the table. -The df values for the denominator of F are printed in a column on the left hand side.
The Null hypothesis and the Independent-Measures T Statistic
*Goal is to evaluate the mean difference between two populations (or between two separate conditions) -Use subscripts to keep track of terms (Mu1=first population mean) (Mu2= Second population mean) -The difference between means is simply Mu1-Mu2 State the Null hypothesis: there is no change, no effect or no difference
Where is the critical region for the Independent measures hypothesis test?
*t distributions vary based on the df -t distribution df =14 Reject at t= +/-2.145 with t=0 in the middle of the distribution
The t statistic for a repeated-measures Research Design
-The single sample t statistic formula will be used to develop the repeated measures t test. t=M-Mu/Sm -The sample mean, M, is calculated from the data, and the value for the population mean, Mu, is obtained from the null hypothesis. -The estimated standard error, Sm, is calculated from the data and provides a measure of how much difference can be expected between a sample mean and the population mean -We are building on the original t score.
Effect size and confidence intervals for the independent measures t statistic
1. Compute the test statistic 2. Make a decision: If the t statistic ratio indicates that the obtained difference between sample means (numerator) is substantially greater than the difference expected by chance (denominator) *We reject H0 (null) and conclude that there is a real mean difference between the two populations or treatments.
Comparing repeated and independent measures designs
A repeated measures design typically requires fewer subjects than an independent-measures design. -The repeated measures design is especially well suited for studying learning, development, or other changes that take place over time. *You can recruit half the participants. *Know for the exam -The primary advantage of a repeated-measures design is that it reduces or eliminates problems caused by individual differences -individual differences are characteristics such as age, IQ, gender, and personality that vary from one individual to another -These individual differences can influence the scores obtained in a research study, and they can affect the outcome of a hypothesis test. -
Independent Measures ANOVA, the F-Ratio has the following structure:
F-Ratio=Variance between treatments/ variance within treatments *For ANOVA, the denominator of the F-Ratio is called the error term. *ANOVA cancels out the errors/ controlling for it. IF there is a meaningful error then your numerator will be very large (larger than the denominator).
analysis of the sum of squares
First, compute a total sum of squares and then partition this value into two components: Between treatments and within treatments. -SS total=sum of squares for the entire set of N scores -It is usually easiest to calculate SS total using the computational formula: SS total= SigmaXsquared -(sigmaX)squared for N. 1. Within Treatments sum of squares is simply the sum of all the SS's within each of the 3 treatment conditions SS within treatments= SigmaSS inside each treatment 2. Between treatments sum of squares is given by... SSbetween= SStotal-SSwithin
The F-Ratio for the repeated-measures ANOVA
Has the same structure that was used for the independent-measures ANOVA. -We are comparing what we actually found with the amount of difference that would be expected if there were no treatment effect.
The Logic of the Repeated-Measures ANOVA
Logically, any differences that are found between treatments can be explained by only two factors 1. Systematic Differences caused by the treatments 2. Random, unsystematic differences *The denominator reflects how much difference/variance is reasonable to expect from random and unsystematic factors. *We want to compare the differences we ACTUALLY found with how much difference is reasonable to expect if there is genuinely no treatment effect (i.e. the null hypothesis is true) -As a result, the repeated measures F-Ratio has the following structure: F= Treatment effects + Random unsystematic differences/ random unsystematic differences
Calculation of the variances (MS values) and the F-Ratio
MS between-treatments= SS between-treatments/ df between-treatments MS error= SS error/ df error F= MS between-treatments/ MS error
Factors in the outcome of a repeated-measures ANOVA/ Measures of Effect Size
Removing individual differences is an advantage only when the treatment effects are reasonably consistent for all the participants. -If the treatment effects are not consistent across participants, the individual differences tend to disappear and value in the denominator is not noticeably reduced by removing them. *This advantage only exists when treatment effects are relatively similar across participants.
Differences scores: The data for a repeated measures study
The difference score for each individual is computed by difference score= D= X2-X1 *Where X1 is the persons score in the first treatment and X2 is the score in the second treatment. *Reaction time measurements taken before and after taking an over the counter medication table 11.2 *Note that MD is the mean for the sample of D Scores.
ANOVA notation N
Total number of scores in the entire study
The Formulas for an independent measures Hypothesis Test (t-test)
We are using the difference between two sample means to evaluate a hypothesis about the difference between two population means. Thus, the independent-measures t formula is t= sample mean difference -population mean difference/estimated standard error =(M1-M2)-(Mu1-Mu2)/S(m1-M2)
Between-Subjects Design/Independent Measures Design
When the two sets of data come from two completely separate groups of participants -CBT + Mindfulness vs CBT -Shock with error vs no shock with error -GAD + MDD vs GAD *This is an independent measures design
within-subjects design
When two sets of data come from the same group of participants. -startle reflex to unpleasant vs. neutral images -pre and post therapy symptoms
The relationship between ANOVA and T-Tests
When you are evaluating the mean difference from an independent measures study comparing only two treatments (two separate samples), you can use either an independent measures t-test or the ANOVA. -the basic relationship between t-statistics and F-ratios can be stated in an equation: F=Tsquared -You will be testing the same hypotheses whether you choose a t-test or an ANOVA. H0=Mu1=Mu2 H1=Mu1 doesn't equal Mu2 The degrees of freedom for the T statistic and the df for the denominator of the F-Ratio (dfwithin) are identical. -The distribution of T and the distribution of F-ratios match perfectly if you take into consideration the relationship F=tsquared
Pairwise comparisons
a post hoc test. The process of conducting pairwise comparisons involves performing a series of separate hypotheses tests. -as you do more and more separate tests, the risk of a type I error accumulates and is called the experimentwise alpha level.
Sm
calculated from the data
Matched subjects Design
each individual in one sample is matched with an individual in the other sample. ie. age, intelligence, SES, Psychopathology, etc. -This is more rare. You recruit 2 groups of people and you make their data related to each other by matching them on some applicable responses (comparing the same IQ or responses). Intentionally relating data to each other. This is not a true experimental design.
Partitioning The Sum of Squares (SS)
for the independent-measures ANOVA SS Total=SigmaXsquared- Gsquared/N 1. SS between treatments n (SS for the treatment means) OR Sigma (Tsquared/n - Gsquared/N) 2. SS within treatments- SigmaSS inside each treatment
Measuring effect size for ANOVA
A significant mean difference simply indicates that the difference observed in the sample data is very unlikely to have occurred just by chance. -Thus, the term significant does not necessarily mean large, it simply means larger than expected by chance. -To provide an indication of how large the effect actually is, it is recommended that researchers report a measure of effect size in addition to the measure of significance. *For ANOVA, the simplest and most direct way to measure effect size is to compute the percentage of variance accounted for by the treatment conditions. -The calculation and the concept of the percentage of variance is extremely straightforward. -Specifically, we determine how much of the total SS is accounted for by the SSbetweentreatments= SSbetweentreatments/SStotal
The t statistic for a repeated measures design
structurally similar to the other t statistics we have examined. -The major distinction of the related samples t is that it is based on difference scores rather than raw scores (X values) *a set of related scores and the differences between them
Complete formula for the independent measures t statistic
t= (M1-M2)-(Mu1-Mu2)/S(m1-m2) t= Sample mean difference - Population mean difference/estimated standard error
Repeated Measures ANOVA and repeated measures T
*The two tests always reach the same conclusion about the null hypothesis. *The basic relationship between the two test statistics is F= tsquared *The df value for the t-statistic is identical to the df value for the denominator of the F-Ratio. *If you square the critical value for the two-tailed t-test, you will obtain the critical value for the F-Ratio. *useful for 2+ groups but usually 3+ in practice *When we compare a t-test to an anova, if we have two groups the results would be the same whether you used a t-statistic or an F statistic (exam).
How to read an F distribution Table
.05 Not bold .01 Bold Numerator is the horizontal Column and Denominator is the vertical column.
Hypothesis Testing and effect size with the repeated measures ANOVA
1. In the first stage of repeated measures ANOVA, the total variance is partitioned into two components: Between-Treatments variance and within-treatments variance. 2. In the second stage, we begin with the variance within treatments and then measure and subtract out the between-subject variance, which measures the size of the individual differences *The remaining variance, often called the residual variance or error variance, provides a measure of how much variance is reasonable to expect after the treatment effects and individual differences have been removed.
A typical situation in which an ANOVA would be used
1. Population 1 (Treatment 1) with resultant sample data of n, M, SS. 2. Population 2 (Treatment 2) with resultant sample data of n, M, SS. 3. Population 3 (Treatment 3) with resultant sample data of n, M, SS. *Omnibus test telling us if there is a difference SOMEWHERE between these groups but it doesn't pinpoint and tell us specifically where that difference is. Bonferroni correction can go in and tell us exactly where that difference is.
Advantages of the Repeated-Measures Design
1. Primary advantages of the repeated-measures ANOVA is the elimination of variability caused by individual differences *In statistical terms, a repeated measures test has more POWER than an independent-measures test; that is, it is more likely to detect a real treatment effect. 2. Sphericity: assumption of sphericity is where the variances of the differences between all combinations of related levels are equal. If you calculated difference scores between different conditions, the variability/variance of those difference scores should be similar or equivalent. a)this assumption is commonly violated. variability of difference scores/the differences between all those conditions. b)Somewhat analogous to homogeneity of variance in a between-subjects ANOVA. There is a correction for this. *If you have four conditions and you run the differences between all of them, you look at the variability of those paired difference scores. (Variability of the scores between 1&2, 2&3, 3&4, 4&1, etc. Advantage: Gives you more statistical power b/c you can capture individual differences. Same set of participants in each condition. You are basically using participants as their own comparison group. B/c we can model mathematically the individual differences that exist, it gives more of an ability to detect a real effect. You can take away the noise variability (between-subject traits).
Assumptions underlying the independent measures t formula
1. The observations within each sample must be independent 2. The two populations from which the samples are selected must be normal 3. The two populations from which the samples are selected must have equal variances
When comparing the F-Ratio Repeated-Measures ANOVA to Independent Measures Designs...
1. The structure of the F-Ratio is the same 2. BUT individual differences are a part of the independent measures F-Ratio but are eliminated from the repeated measures F-Ratio.
The T statistic for ANOVA
1. The test statistic for ANOVA is very similar to the t statistics used in earlier chapters. -For the t statistic, we first computed the standard error, which measures the difference between two sample means that is reasonable to expect if there is no treatment effect (that is, if H0 is true). *conceptually, the t test and ANOVA is set up to do similar things. T-tests compare two means which just gets more complex in an ANOVA. 2. For the ANOVA, we want to compare the differences among two or more sample means. -with more than 2 samples, the concept of "difference between sample means" becomes difficult to define or measure. *The solution to this problem is to use variance to define and measure the size of the differences among the sample means. -Variance is a way to operationalize the spread between scores. In an ANOVA we capitalize on what variance does. IF variance is large enough across sample means that gives us an idea of group differences.
What two interpretations do we have to decide between in an ANOVA?
1. There really is no difference between the populations or treatments. The observed differences between the sample means are caused by random, unsystematic factors (sampling error). ERROR 2. The populations or treatments really do have different means, and these population mean differences are responsible for causing systematic differences between the sample means. THERE IS A MEANINGFUL DIFFERENCE IN TREATMENT GROUPS. Any discrepancies are because they are meaningfully different from one another. *ANOVA= OMNIBUS test
To develop the formula for S(m1-m2) *calculating the estimated standard error* we consider which 3 points?
1. We assume error between the sample means and population means 2. The amount of error associated with each sample mean is measured by the estimated standard error of M. (meaning we can calculate it!) 3.For the independent measure T statistic, we want to know THE TOTAL AMOUNT OF ERROR involved in using two estimates (sample means) of the population parameters (population means) a. to do this, if the samples are the same size we will find the error from each sample separately and then add the two error together b. When the samples are of different sizes, a pooled or averaged estimate (weighted), that allows the bigger sample to carry more weight in determining the final value is used.
The research designs that are used to obtain the two sets of data for independent measures design can be classified into two general categories
1. Within Subjects Design 2. Between Subjects Design
Repeated Measures Design
A within subjects design= the dependent variable is measured two or more times for each individual in a single sample. -The same group of subjects is used in all of the treatment conditions 1. Affective Neuroscience=passive picture viewing task 2. Cognitive tasks= stroop *You want related groups of people here (not separate groups) The idea is that you are measuring a dependent variable 2+ times (not the IV)
An overview of analysis of variance
Analysis of variance (ANOVA) is a hypothesis-testing procedure that is used to evaluate mean differences between 2+ treatment conditions (or populations).
Repeated measures and matched subjects designs
Because the scores in one set are directly related, one on one, with the scores in the second set, the two research designs are statistically equivalent and share the common name related samples design (or correlated samples design). -Table 11.1 in book -Same set of people with two sets of scores
What is one technique for measuring the effect size in an independent measure t statistic?
Cohens D: produces a standardized measure of mean difference
Type I Errors and Multiple-Hypothesis Tests
Each time you do a hypothesis test, you select an alpha level that determines the risk of a Type I error. - Often a single experiment requires several hypothesis tests to evaluate all the mean differences. *Each test will have a risk of a type I error, and the more tests you do the greater the risk (they add on each other). For this reason, Researchers often make a distinction between the testwise alpha level and the experimentwise alpha level is the total probability of a type I error that is accumulated from all of the individual tests in the experiment. *ANOVA's allow us to look at individual differences. Combined risk (increased risk of Type I error) due to running so many analyses. *Start out with an ANOVA and follow up with a t-test if there is a difference found. ANOVA reduces the amount of analyses you need to run initially (5+ groups is very helpful). We are trying to minimize type I errors.
What is the final F-Ratio?
F= Variance/difference between treatments without individual differences (numerator)/ Variance/differences with no treatment effect with individual differences removed (denominator)
The F-Ratio: The Test Statistic for ANOVA
For the Independent Measures ANOVA, the F-Ratio has the following structure F-Ratio= variance between treatments/ variance within treatments. *if the null hypothesis is true, we expect F to be about 1.0 *if the null hypothesis is false, F should be much greater than 1.00
The F-Ratio: The Test Statistic for ANOVA
For the independent-measures ANOVA, The F-Ratio has the following structure: F-Ratio=Variance between treatments/Variance within treatments Between-Treatments Variance= Measures differences caused by systematic treatment effects and random, unsystematic factors Within-Treatment Variance= Measures differences caused by Random, unsystematic factors. *Random error is in the denominator.
What is stage 1 of the repeated measures analysis
Identical to the independent-measures ANOVA. -SStotal=SigmaXsquared-Gsquared/N dftotal=N-1 SSwithin-treatments= SigmaSSinside each treatment dfwithin-treatments=Sigmadfinside each treatment SSbetween-treatments=Sigma(Tsquared/n)- (Gsquared/N) dfbetween-treatments=k-1
Calculation of variances (MS) and the F-Ratio
In ANOVA, it is customary to use the term MEAN SQUARE or simply MS in place of the term variance. -For the final F-Ratio, we will need an MS (Variance) between treatments for the numerator and an MS (variance) within treatments for the denominator. In each case, MS(variance)=ssquared= SS/df MSbetween= Squaredbetween= SSbetween/dfbetween MSwithin=Ssquaredwithin= SSwithin/dfwithin
Hypothesis tests for the repeated-Measure design
In a repeated measures study, each individual is measured in two different treatment conditions and we are interested in whether there is a systematic difference between the scores in the first treatment condition and the scores in the second treatment condition. -A difference score is computed for each person -They hypothesis test uses the difference scores from the sample to evaluate the overall mean difference, MuD, for the entire population. -Two different treatment conditions -Interested in difference scores not in their scores in isolation for the first/second condition. We just want the contrast testing population level hypothesis against the sample.
Factor
In an Analysis of Variance, ANOVA, the variable (either independent or quasi-independent) that designates the groups being compared is the factor. -The individual conditions or values that make up a factor are called the LEVELS of the factor -A study that combines two factors is called a two-factor design or a factorial design. Factor= variable Within each factor is a level or condition
Analysis of Degrees of Freedom
In computing the df, there are two important considerations to keep in mind: 1. Each df value is associated with a specific SS value 2. Normally, the value of df is obtained by counting the number of items that were used to calculate SS and then subtracting 1. df=n-1 *To find the df associated with SStotal remember -the SS value measures variability for the entire set of N scores. dftotal= N-1 3. To find the df associated with SSwithin, we must look at how this SS value is computed. -Remember, we first find SS inside of each of the treatments and then add these values together. -Each of the treatment SS values measures variable for the n scores in the treatment, so each SS has df=n-1. When all these individual treatment values are added together, we obtain; dfwithin=Sigma(n-1)= Sigmadf in each treatment *Notice that the formula for dfwithin simply adds up the number of scores in each treatment (the n values) and subtracts 1 for each treatment. If these two stages are done separately, you obtain dfwithin=N-k 4. The df associated with SSbetween can be found by considering how the SS value is obtained. -This SS formula measures the variability for the set of treatments (totals or means). To find dfbetween, simply count the number of treatments and subtract 1. Because the number of treatments is specified by the letter K, the formula for df is dfbetween=k-1.
Statistical Hypotheses for ANOVA
In general, H0 states that there is no treatment effect. In an ANOVA with three groups H0 could appear as... H0=Mu1=Mu2=Mu3... The Alternative Hypothesis H1 states that the population means are not all the same. H1= There is at least one main difference. (No symbols) Just an omnibus test to identify that a difference exists SOMEWHERE in the groups. *Math is not more difficult there are just more steps to take.
Directional Hypotheses and one-tailed tests
In many repeated measures and matched subject studies, the researcher has a specific prediction concerning the direction of the treatment effect. -This kind of directional prediction can be incorporated into the statement of the hypotheses, resulting in a directional or one-tailed, hypothesis test.
Within-Treatment Variance
Inside each treatment condition, we have a set of individuals who receive the same treatment. -The researcher does not do anything that would cause these individuals to have different scores, yet they usually do have different scores. -The differences represent random and unsystematic differences that occur when there are no treatment effects. -Thus, the within-treatments variance provides a measure of how big the differences are when H0 is true. *Variability has nothing to do with the researcher. We use that as a measure of error. The kind of variability you would expect if the null hypothesis were true.
Major advantage of ANOVA's
It can be used to compare 2+ treatments. Limitation of a t-test statistic is that they can only be used for 2 conditions or groups at the same time.
What is the main advantage of a repeated measured study?
It uses exactly the same individuals in all treatment conditions -there is no risk that the participants in one treatment are substantially different from the participants in another (your comparison group is the same as your experimental group). -Participants are their own control group.
Post Hoc Tests
Post-tests. Additional hypothesis tests that are done after an ANOVA to determine exactly which mean differences are significant and which are not. In statistical terms, this is called making pairwise comparisons.
ANOVA notation T
SigmaX (sum of scores) for each treatment condition (The Treatment Total)
What does the standard error measure in each of the t-score formulas?
Standard error is the denominator of the t-score formula and it measures how accurately the sample statistic represents the population parameter. -In the single sample t formula, the standard error measures the amount of error expected for a sample mean and is represented by Sm. *For the independent-measures t formula, the standard error measures the AMOUNT OF ERROR that is expected when 1. You use a sample mean difference (M1-M2) to represent a population mean difference (Mu1-Mu2) 2. The standard error for the sample mean difference is represented by the symbol S(m1-m2)
The partitioning of Variance in a repeated measures study
Start with your total variance. Stage 1: A) This is your numerator of the F-Ratio. Between-Treatments Variance. Your treatment effect, and error/chance (excluding individual differences). B)Within Treatment Variance is your individual differences and other errors that can occur. Stage 2: A) This is your denominator of the F-Ratio. Error Variance: error excluding individual differences. B)Between subjects variance is your individual differences.
Steps for figuring out an ANOVA
Step1: Find the SS total for between and within. FInd the df total for between and within Step2: Each variance in the F-Ratio is computed as SS/df Variance between treatments: SS Between/df Between Variance Within Treatments: SS within/df Within Step 3: The final goal for the ANOVA is an F-Ratio. F= Variance Between treatments/Variance within treatments F= Variance Between Treatments/ Variance within Treatments
Hypothesis tests with the independent measures T statistic
The Independent Measures T statistic uses the data from two separate samples to help decide whether there is a significant mean difference between two populations (or between two treatment conditions). 1. state the hypothesis and select the alpha level 2. Compute the df for an independent-measures design 3. Obtain the data and compute the test statistic 4. Make a decision
The Scheffe Test
The Scheffe Test uses an f-ratio to evaluate the significance of the difference between any two treatment conditions. -One of the most conservative posthoc tests (smallest risk of a type I error) -The numerator of the F ratio is an MS between treatments that is calculated using only the two treatments you want to compare -The denominator is the same MSwithin that was used for the overall ANOVA. *The safety factor for the Scheffe test comes from the following two considerations: 1. The Scheffe test uses the value of K from the original experiment to compute df between treatments. Thus df for the numerator of the F-ratio is k-1 2. The critical value is the same as was used to evaluate the F-ratio from the overall ANOVA.
Assumptions of the repeated measures ANOVA
The basic assumptions for the repeated-measures ANOVA are relatively similar to those required for the independent-measures ANOVA. 1. Each person should have their own unique scores. The observations within each treatment condition must be independent. 2. The population distribution within each treatment must be normal. 3. Variances surrounding population distributions should be relatively similar. The variances of the population distributions for each treatment should be equivalent. 4. Assumption of Sphericity: Where the variances of the differences between all combinations of related levels are equal -commonly violated -somewhat analogous to homogeneity of variance in a between-subjects AVOVA.
Between-Treatments Variance
The between-Treatments variance simply measures how much difference exists between the treatment conditions. -There are two possible explanations for these between-treatment differences 1. The differences are the result of sampling error 2. The differences between treatments have been caused by the treatment effects *Some variability due to sampling error or from the treatment effect (Your intervention is working).
The estimated standard error of M1-M2 in an independent measure hypothesis T-Test can be interpreted two ways
The estimated standard error of M1-M2 can be interpreted in two ways. 1. The standard error is defined as a measure of the standard or average distance between a sample statistic (M1-M2) and the corresponding population parameter (mu1-mu2). 2*So what does that mean? When the null hypothesis is true, the standard error is measuring how big, on average, the sample mean difference is.
ANOVA Formulas
The final calculation for ANOVA is the F-Ratio, which is composed of two variances: F-Ratio= variance between treatments/variance within treatments. *Each of the two variances in the F-Ratio is calculated using the basic formula for sample variance. Sample variance= s-squared= SS/df
Assumptions for the Independent-Measures ANOVA
The independent measures ANOVA requires the same three assumptions that were necessary for the independent measures t hypothesis test 1. The observations within each sample must independent. 2. The populations from which the samples are selected must be normal 3. The populations from which the samples are selected must have equal variances (homogeneity of variance). *ANOVA is considered a relatively robust analysis. -can tolerate violating the homogeneity of variance assumption relatively well *The assumption of homogeneity of variance is an important one. If a researcher suspects it has been violated, it can be tested by Hartley's F-max test for homogeneity of variance.
The hypotheses for the repeated measures ANOVA's are the same as...
The independent measures. The Null Hypothesis states that for the general population there is no mean differences among the treatment conditions being compared. In symbols: H0= Mu1=Mu2=Mu3=.... The Alternative Hypothesis states that there ARE mean differences among the treatment conditions. (Rather than specifying exactly which treatments are different, we use a generic version of H1, which simply states that differences exist.
What is the repeated measures ANOVA used to evaluate?
The mean differences in two general research situations. 1. An experimental study in which the researcher manipulates the independent variable to create 2+ treatment conditions, with the same group of individuals tested in all of the conditions. 2. A nonexperimental study in which the same group of individuals is simply observed at two or more different times.
Measuring Effect Size for the repeated-measures analysis of variance
The most common method for measuring effect size with ANOVA is to compute the percentage of variance that is explained by the treatment differences. -The formula for computing effect size for a repeated-measures ANOVA is edasubscriptPsquared= SS between-treatments/ SS total- SS between-subjects =SS between-treatments/ SS between-treatments + SSerror *This is a measure of overall effect size. This gives an idea of the entire condition (not like CohensD). This doesn't give specific group comparison effect size conditions (specific conditions).
Effect size and confidence intervals for the repeated-Measures T Test
The most commonly used measures of effect size are cohens D. and r-squared, the percentage of variance accounted for. Estimated D= M subD IS *specific and tailored methodology just for the repeated measures T-test *The size of the treatment effect also can be described with a confidence interval estimating the population mean difference, MuD. MuD=MD +/-tsMd
Time related factors and order effects
The primary disadvantage of a repeated-measures design is that the structure of the design allows for factors other than the treatment effect to cause a participants score to change from one treatment to the next. -Specifically, in a repeated-measures design, each individual is measured in two different treatment conditions, often at two different times. *One way to deal with time-related factors and order effects is to COUNTERBALANCE the order of presentation of treatments. -That is. the participants are randomly divided into two groups, with one group receiving treatment one followed by treatment two. The other group receiving treatment two followed by treatment one. -The goal of counterbalancing is to distribute any outside effects evenly over the two treatments. Disadvantage of within: There are learning effects that can happen. ie. mock jury making. Don't let the participants learn over time.
Assumptions of the Related Samples T Test
The related samples T statistic requires two basic assumptions 1. The observations within each treatment condition must be independent. -Notice that the assumption of independence refers to the scores WITHIN each treatment 2. The population distribution of difference scores (D values) must be normal -Independent: Scores must not be related to each other. *Focus on across conditions independence (not what we are talking about here) for the exam.
The hypotheses for a related Samples Test
The researcher's goal is to use the sample of difference scores to answer questions about the general population. -The researcher would like to know whether there is any difference between the two treatment conditions for the general population. -Between Subjects= interested in group scores. Treated sample vs. untreated sample. -within subjects= same set of people but different conditions. Hoping the pretreatment sample is the same as the posttreatment sample. *For a repeated measures study, the null hypothesis states that the mean difference for the general population is zero. In symbols: H0: MuD=0 *The alternative hypothesis states that there is a treatment effect that causes the scores in one treatment condition to be systematically higher (or lower) than the scores in the other condition. In symbols: H1: MuD doesn't equal 0 -Population Mean Mu with the added D if the difference scores you are talking about. Two condition differences (pre and post tests)
ANOVA notation G
The sum of the scores in the research study (The Grand Total)
Conceptual Test statistic for ANOVA: Setting up the ratio
The test statistic for the ANOVA uses the fact that differences between sample means is too hard to define for 3+ groups (so we use variance to define and measure the size of the differences among the sample means) to compute an F-Ratio with the following structure. F= Variance between sample means/ variance expected with no treatment effect *The analysis process divides the total variability into two basic components 1. Between-Treatments variance 2. Within-Treatment Variance *Divides the variabilities
Homogeneity of variance
The third assumption is referred to as homogeneity of variance and states that the two populations being compared must have the same variance. - This is particularly important when you have large sample size differences
Null Hypothesis for the independent measure t test
There is no change, no effect, or no difference. H0=Mu1-Mu2=0 The alternative hypothesis then is H1= Mu1-Mu2 does not equal 0
The Distribution of F-Ratios
To determine whether we reject the null hypothesis, we have to look at the distribution of F-Ratios Two obvious characteristics: 1. F values always are positive numbers because variance is always positive. 2. When H0 is true, the numerator and denominator of the F-ratio are measuring the same variance. Then Sketch the distribution of F-Ratios 1. The distribution is cut off at zero (all positive values), piles up around 1.00, and then tapers off to the right 2. The exact shape of the F distribution depends on the df for the two variances in the F ratio
Tukey's Honestly Significant Difference (HSD) Test
Tukey's test allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance. This value, called the honestly significant difference (HSD) is then used to compare any two treatment conditions. -If the mean difference exceeds Tukey's HSD, you conclude that there is a significant difference between the treatments -Otherwise you cannot conclude that the treatments are significantly different.
The role of sample variance and sample size in the Independent Measures T statistic
Two factors that play important roles in the outcomes of hypothesis tests are the VARIABILITY OF THE SCORES and the SIZE OF THE SAMPLES. -Both factors influence the magnitude of the estimated standard error in the denominator of the t statistic *The standard error is directly related to sample variance so that larger variance leads to larger error. -As a result, larger variance produces a smaller value for the t statistic (closer to zero) and reduces the likelihood of finding a significant result *By contrast, the standard error is inversely related to sample size (larger size leads to smaller error) -Thus, a larger sample produces a larger value for the t statistic (farther from zero) and increases the likelihood of rejecting H0 (Null).
Affective Neuroscience
Within subjects differences between two or more experimental conditions are used as an individual difference measure -Fear acquisition -Threat sensitivity -Emotional Processing and regulation -Reward Sensitivity *these individual differences are robustly observed but there are increased concerns about reliability. Effects consistent over time? Across trials? Reliability is a necessary but not sufficient condition for validity. *Often times studies are interested in between-group differences using repeated measures design. -MDD v. no MDD in reward sensitivity -High/low psychopathy in threat sensitivity -Increasing focus in psychological science (NIMH RDoC
Partitioning the Degrees of Freedom (df)
dftotal N-1 dfbetweentreatments k-1 dfwithintreatments Sigma(n-1)=N-k *Partitioning the df for the independent measures ANOVA
For the repeated measures design, the sample data are...
difference scores and are identified by the letter D, rather than X. -The population mean that is of interest to us is the population mean difference (the mean amount of change for the entire population) and we identify this parameter with the symbol MuD -Difference scores not individual scores -We are looking at contrast between conditions. Repeated sample T test: What population has a meaningful difference depending on what condition they are in. (same population)
ANOVA notation K
identifies the number of treatment conditions (the number of level of factors *For an independent-measures study, k also specifies the number of separate samples.
ANOVA notation and formulas
k=the number of treatment conditions: The number of levels of the factor n=the number of scores in each treatment N= the total number of scores in the entire Study T=SigmaX=The sum of scores for each treatment condition (Treatment total) G=The sum of all the scores in the research study (The Grand Total)
What is stage 2 of the repeated measures analysis?
measure the individual differences and then remove them from the denominator of the F-Ratio SSbetween-subjects=Sigma(Psquared/k)- (Gsquared/N) SSerror=SSwithin-treatments- SSbetween-subjects dfbetween-subjects= n-1 dferror= dfwithintreatments- dfbetween-subjects *Between subjects variability is being measured. *all the variability within those treatment conditions and taking out the individual differences (between subjects variability). This leaves us error left over.
Mu
population mean generated from the null hypothesis is 0.
With the simple changes, the t formula for the repeated-measures design becomes...
t=MD-MuD/SMD -In this formula, the estimated standard error, SMD, is computed in exactly the same way as it is computed for the single-sample t statistic. -The first step is to compute the variance (or the standard deviation) for the sample of D scores. s-squared=SS/n-1 * Mean difference minus population difference/ standard error of these differences SMD=1 Compute the variance (or SD)SMD= square root of s-squared/n *with these simple changes, the t formula for the repeated measures design becomes t=MD-MuD/SMD -In this formula, the estimated standard error, SMD, is computed in exactly the same way as it is computed for the single-sample t statistic. (the estimated standard error is then computed using the sample variance and the sample size n). SMD= Square root of S-square/n
df for the t statistic
the degrees of freedom for the Independent measures T statistic are determined by the df values for the two separate samples df for the t statistic= (n1-1)+(n2-1) =n1+n2-2
Confidence Intervals for the Independent Measures t statistic
we use a sample mean difference M1-M2 to estimate the population mean difference Mu1-Mu2 1. The first step is to solve the t equation for the unknown parameter. For the independent measures t statistic we obtain mu1-mu2=M1-M2+ts(m1-m2) Sample 1: M1=12 Sample 2: M2=8 S(m1-m2):=1.5 df=14 *In addition to describing the size of a treatment effect, estimation can be used to get an indication of the significance of the effect. *Looking for a 95% confidence interval
