PSY201: Chapter 10 - Independent Measures Flashcards
Confidence Intervals revisited
Therangeofscoreslikelytoincludethetruepopulationmean n Tells us how confident we are in our findings (based on the
chosen alpha level)
q e.g.,we’re95%confidentoursamplemeanfallsbetweena range of values
M +/- (t critical)(SM)
Confidence Intervals: when useful
n t critical = 2.16
n Whatifift(13)=2.18?
Independent-Measures Designs
The independent-measures hypothesis test (or between-subjects design) expands the single-sample t test to test for differences between two
separate groups
n Compares differences in the distribution between means
Independent-Measures Designs
q One from one population, the other from a second population
n Allows researchers to evaluate the mean difference between two
populations using the data from two separate samples.
Independent-Measures Designs
n Can be used to test for mean differences between two distinct populations (e.g., men vs. women) or between one sample given two different treatment conditions (e.g., drug vs. no-drug).
Independent-Measures Design examples
A Social psychologist in interested in developmental trends in aggression. She compares the type and amount of aggressive behaviour in a group of 11 year old girls to those in a group of 11 year old boys (n = 50 in each group).
n Do Canadian young adults have different opinions about ethnic diversity than Canadian older adults?
n What is the effect of 0 mg vs 10 mg of a particular drug on reducing seizures in rats?
Independent-Measures Design examples
Which is more effective, cognitive behaviour therapy (CBT) or drug therapy for treating depression in young teenagers?
In these cases, we may not know anything about the parent population distributions, except that if our samples are representative, the parent populations probably look a lot like our samples, and vice versa
Independent-Measures Designs
The independent-measures design is used in situations where a researcher has no prior knowledge about either of the two populations (or treatments) being compared.
n In particular, the population means and standard deviations are all unknown.
Independent-Measures Designs
Because the population variances are not known, these values must be estimated from the sample data.
Hypothesis Testing with the Independent- Measures t Statistic
As with all hypothesis tests, the general purpose of the independent-measures t test is to determine whether the sample mean difference obtained in a research study indicates a real mean difference between the two populations (or treatments) or whether the obtained difference is simply the result of sampling error.
Hypothesis Testing with the Independent- Measures t Statistic
The only difference between the t formula and the z-score formula is that the z-score uses the actual population variance, σ2 (or standard deviation), and the t formula uses the corresponding sample variance S (or standard deviation) when the population value is not known”
Hypothesis Testing with the Independent- Measures t Statistic
For the independent t, G & W point out that it’s is technically a test of difference between sample mean difference and the hypothesized population mean difference …
Independent t Single sample (formula)
Hypothesis Testing with the Independent- Measures t Statistic
Same logic behind hypothesis testing applied as with one sample
n However, important differences with two groups
q Incorporate2populationmeansintohypotheses
q Look at hypothetical distributions of sample mean differences when testing hypotheses
Hypothesis Testing with the Independent- Measures t Statistic
Thus, estimated standard error of the mean includes 2 measures of variability and 2 sample sizes
q Degrees of freedom include the n of each group
Hypothesis Testing with the Independent- Measures t Statistic
To prepare the data for analysis, the first step is to compute the sample mean and SS (or s, or s2) for each of the two samples.
n The hypothesis test follows the same four-step procedure outlined in Chapters 8 and 9.
Hypothesis Testing with the Independent- Measures t Statistic
- State the hypotheses and select an α level. For the independent-measures test, H0 states that there is no difference between the two population means.
Hypothesis Testing with the Independent- Measures t Statistic
- Locate the critical region. The critical values for the t statistic are obtained using degrees of freedom that are determined by adding together the df value for the first sample and the df value for the second sample. You then subtract two instead of one to reflect the two groups.
Hypothesis Testing with the Independent- Measures t Statistic
- Compute the test statistic. The t statistic for the independent- measures design has the same structure as the single sample t introduced in Chapter 9. However, in the independent-measures situation, all components of the t formula are doubled: 2 sample means, 2 population means, and 2 sources of error contributing to the standard error in the denominator.
Table Slide 14
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Hypothesis Testing with the Independent- Measures t Statistic
In order to calculate an independent samples t statistic, you’ll need to determine the difference between sample means and estimate the error term (estimated SEM)
q Two sources of error contributing to the standard error
in the denominator»_space;
n To estimate the error term (s(M1-M2) in your text) you’ll need to know how to calculate pooled variance or pooled standard deviation
Hypothesis Testing with the Independent- Measures t Statistic
q Pooled variance – a weighted average of the variance in each sample distribution
n Works for both = and ≠ size samples
n We assume samples come from populations with equal variances
Hypothesis Testing with the Independent- Measures t Statistic
Given that our error term is typically defined as a standard deviation value divided by √n, it’s easy to see how to use the pooled variance or pooled standard deviation to estimate SEM (formula)
Hypothesis Testing with the Independent- Measures t Statistic
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 and conclude that there is a real mean difference between the two populations or treatments.
Hypothesis Testing with the Independent- Measures t Statistic
Note re: the t-table
q Whentargetdfavailableintable,noproblem
q When not available, need to estimate
Hypothesis Testing with the Independent- Measures t Statistic
q Alwayserronthesideofconductingamoreratherthanless conservative test
q Iftargetdffallsbetweenavailabledfvalues,chooselowerdf (larger t critical value)
Homogeneity of Variance Assumption
Althoughmosthypothesistestsarebuiltonasetofunderlying assumptions, the tests (including t tests) usually work reasonably well even if the assumptions are violated.
n Theoneimportantexceptionistheassumptionofhomogeneityof variance for the independent-measures t test.
Homogeneity of Variance Assumption
n The assumption requires that the two populations from which the samples are obtained have equal variances.
n Thisassumptionisnecessarytojustifypoolingthetwosample variances and using the pooled variance in the calculation of the t statistic.
Homogeneity of Variance Assumption
Before you can average (or pool) the two sample variances it is necessary that both samples are estimating the same population variance. See text for excellent discussion of this
Homogeneity of Variance Assumption
Iftheassumptionisviolated,thenthetstatisticcontainstwo questionable values: (1) the value for the population mean difference which comes from the null hypothesis, and (2) the value for the pooled variance.
Homogeneity of Variance Assumption
The problem is that you cannot determine which of these two values is responsible for a t statistic that falls in the critical region.
n In particular, you cannot be certain that rejecting the null hypothesis is correct when you obtain an extreme value for t.
Homogeneity of Variance Assumption
If the two sample variances appear to be substantially different, you should use Hartley’s F-max test (sometimes called Levene’s test) to determine whether or not the homogeneity assumption is satisfied.
F-max = largest sample variance/smallest sample variance
s2 (largest)/s2 (smallest)
Homogeneity of Variance Assumption
Compare with critical value (Table B.3)
n Violation of homogeneity assumption:
q F-max value > critical value
n If homogeneity of variance is violated, there is an alternative procedure that does not involve pooling the two sample variances.
Measuring Effect Size for the Independent-Measures t
n Effectsizefortheindependent-measurestismeasured in mostly the same way that we measured effect size for the single-sample t
q estimate of Cohen’s d measures the size of the treatment effect in terms of the standard deviation
= Estimated mean difference / estimated standard deviation = M1 – M2 / √S2p
Measuring Effect Size for the Independent-Measures t
q r2toobtainameasureofthepercentageofvarianceaccounted for by the treatment effect.
(formula)
Measuring Effect Size for the Independent-Measures t
By measuring the amount of variability that can be attributed to the treatment, we obtain a measure of the size of the treatment effect. For the t statistic hypothesis test,
Measuring Effect Size for the Independent-Measures t
t2 + df
percentage of variance accounted for =
r2 = ─────
Summary of basic steps in hypothesis testing
n Identify IV DV & basic question being asked
n Establish hypotheses and alpha level
n Examine assumptions
q The data in each group is normally distributed
n descriptive statistics and graphs can help with this
q Homogeneity of variance (the variance in each is similar)
Summary of basic steps in hypothesis testing
n Choose appropriate test to use
n Calculate test statistic and conduct significance test
n Calculateeffectsize(s)
n Write up and Interpret findings
q Conduct any other tests that seem relevant
Describing Your Results Visually
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Write up findings
An independent measures t test was conducted. As shown in Figure 1, a significant difference was found between eighth (M = 90.76, SD= 19.32) and tenth (M = 99.32, SD = 18.36) grade students in the interpretation of musical meaning, t(398) = 4.55, p < 0.05, d = 0.45. According to Cohen’s guidelines, this is a medium effect size
Presenting results of the Independent t
test
independent measures t test found that the individuals under stress (M = 5.87, SD = 1.06) recalled significantly less digits than those not under stress (M = 6.80, SD = 1.32), t(28) = -2.13, d = 0.77. According to Cohen’s guidelines, this is a large effect, suggesting that stress can quite badly impair memory
