The t Test for Dependent Means (not single sample - two sample)

What is an assumption of the t test?

- A normal population distribution

What can power be expressed as?

- A probability or a percentage

Power for a t test for dependent means

- Can be determined using a power table, a power software package, or an Internet power calculator

What can studies that use the t test for dependent means be?

- Extremely sensitive Surprisingly few participants are needed when a researcher can find a way to set up a repeated measures design in which difference scores are the basic unit of analysis.

T-test for dependent means

- Hypothesis-testing procedure in which there are two scores for each person and the population variance is not known. It determines the significance of a hypothesis that is being tested using "difference" or "change scores" from a single group of people Called "dependent means" because the mean for each group of scores are dependent on each other in that they are both from the same people (when you analyze t-scores for independent means, you compare scores from two different groups of people) Also called a paired-samples t test, t test for correlated means, t test for matched samples and t test for matched pairs.

Population 1 and 2

- Population 1 receives the experimental change - Population 2 is the null hypothesis

What about t tests is noted in research articles?

- The degrees of freedom, the t score & the significance level ex: t(24) = 2.80, p < .05 Tells you that the researcher used a t test with 24 degrees of freedom, found a t score of 2.80 and result was significant at the 0.05 level. - Whether a one-tailed or two-tailed test was used may also be noted. Usually the means, and sometimes the standard deviations, are given for each testing. Rarely does an article report the standard deviation of the difference scores. Psychologists rarely use the t test for a single sample. However, one sometimes sees the t tests for a single sample in research articles. T tests for dependent means more commonly used than single sample t tests. Often the results of a t test for dependent means will be given in the text and not in a table.

Figuring effect size for a study using a t test for dependent means

- The difference between the population means divided by the population standard deviation - u1 is for the predicted mean of the population of difference scores - u2 is almost always 0 - o stands for the standard deviation of the population of difference scores Conventions for effect size are the same as for Z-tests (small, medium and large). - Can estimate effect size before or after a study

What is a condition for the comparison distribution being a t distribution?

- The distribution of individuals following a normal curve. If the distribution of individuals does not follow a normal curve (if it is not a 'normal' population), the comparison distribution will follow some other (usually unknown) shape.

What is a common situation in which using a t test for dependent means is likely to give a seriously distorted result?

- When using a one-tailed test & the population is highly skewed (very asymmetrical, with a much longer tail on one side than the other).

What do you not know when doing a t test?

- Whether the population is normal (because when doing a t test usually you only know the scores in your sample) However, distributions in psychology research quite often approximate a normal curve. Also, statisticians have found that, in practice, you get reasonably accurate results with t tests even when the population is far from normal -> t test is robust over moderate violations of the assumption of a normal population distribution

What can you use a t test for, other than comparing two sets of scores from the same person?

- You can use scores from pairs of research participants (natural or created pairs) Create difference scores by finding the difference between individual partner's scores

What is true about most research situations?

- You do not know the population's variance AND its mean ALSO have not one, but two sets, of scores

When do you use t tests for a single sample compared to t tests for dependent means?

- You use a t test for a single sample when you know the population mean & you have one score for each participant. - You use the t test for dependent means when you do not know the population mean & there are two scores for each participant.

Outline for Writing Essays for a t Test for a Single Sample 1.

1. Describe the core logic of hypothesis testing in this situation. Be sure to mention that the t test for a single sample is used for hypothesis testing when you have scores for a sample of individuals and you want to compare the mean of this sample to a population for which the mean is known but the variance is unknown. Be sure to explain the meaning of the research hypothesis and the null hypothesis in this situation. 2. Outline the logic of estimating the population variance from the sample scores. Explain the idea of biased and unbiased estimates of the population variance, and describe the formula for estimating the population variance and why it is different from the ordinary variance formula. 3. Describe the comparison distribution (the t distribution) that is used with a t test for a single sample, noting how it is different from a normal curve and why. Explain why a t distribution (as opposed to the normal curve) is used as the comparison distribution. 4. Describe the logic and process for determining the cutoff sample score(s) on the comparison distribution at which the null hypothesis should be rejected. 5. Describe how and why you figure the t score of the sample mean on the comparison distribution. 6. Explain how and why the scores from Steps ❸ and ❹ of the hypothesis-testing process are compared. Explain the meaning of the result of this comparison with regard to the specific research and null hypotheses being tested.

Outline for Writing Essays for a t Test for Dependent Means

1. Describe the core logic of hypothesis testing in this situation. Be sure to mention that the t test for dependent means is used for hypothesis testing when you have two scores from each person in your sample. Be sure to explain the meaning of the research hypothesis and the null hypothesis in this situation. Explain the logic and procedure for creating difference scores. 2. Explain why you use 0 as the mean for the comparison distribution. 3. Outline the logic of estimating the population variance of difference scores from the sample scores. Explain the idea of biased and unbiased estimates of the population variance, and describe the formula for estimating the population variance. Describe how to figure the standard deviation of the distribution of means of difference scores. 4. Describe the comparison distribution (the t distribution) that is used with a t test for dependent means. Explain why a t distribution (as opposed to the normal curve) is used as the comparison distribution. 5. Describe the logic and process for determining the cutoff sample score(s) on the comparison distribution at which the null hypothesis should be rejected. 6. Describe how and why you figure the t score of the sample mean on the comparison distribution. 7. Explain how and why the scores from Steps ❸ and ❹ of the hypothesis-testing process are compared. Explain the meaning of the result of this comparison

Steps for a t Test for Dependent Means

1. Restate the question as a research hypothesis and a null hypothesis about the population 2. Determine the characteristics of the comparison distribution a) Make each person's two scores into a difference score. Do all the remaining steps using these difference scores. b) Figure the mean of the difference scores c) Assume a mean of the distribution of means of difference scores of 0 d) The standard deviation of the distribution of means of difference scores is figured as follows: i) Figure the estimated population variance of difference scores -> S^2 = SS/df ii) Figure the variance of the distribution of means of difference scores -> S^2M = S^2 / N iii) Figure the standard deviation of the distribution of means of difference scores e) The shape is a t distribution with df = N - 1 3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected a) Decide the significance level and whether to use a one-tailed or two-tailedd test b) Look up the appropriate cutoff in a t table. 4. Determine your sample's score on the comparison distribution: t = (M - u)/SM. 5. Decide whether to reject the null hypothesis: Compare the scores from Steps 3 & 4

How does a t test for dependent means differ from the way you do a t test for a single sample?

1. You use difference scores 2. You assume that the population mean (of the difference scores) is 0.

Assumption

Condition, such as a population's having a normal distribution, required for carrying out a particular hypothesis-testing procedure. A part of the mathematical foundation for the accuracy of the tables used in determining cutoff values.

Difference scores

Difference between a person's score on one testing and the same person's score on another testing: Often an after-score minus a before-score, in which case it is also called a change score. - After you have the difference score for each person in the study, you do the rest of the hypothesis testing with difference scores (treat the study as if there were a single sample of scores).

Robustness

Extent to which a particular hypothesis-testing procedure is reasonably accurate even when its assumptions are violated.

What do studies using difference scores often have?

Larger effect sizes (& high power) for the same amount of expected difference between means than other kinds of research designs Testing each of a group of participants twice produces larger power than dividing the participants up into two groups and testing each group once (one group tested under one condition and the other tested under another condition). This is because the standard deviation of difference scores is usually quite low (which is the denominator of effect size). Only difference comes from comparing participants to themselves, where the variation in them is small. On the other hand, testing a group of participants before and after an experimental procedure, without any kind of control group that does not go through the procedure, is a weak research design. Even if study produces a significant difference, it leaves many alternative explanations for that difference.

What does effect size affect?

The amount of participants you need to have 80% power (the minimum power to make a study worth doing).

What is the effect of violating the assumption of a test for dependent means (that the population of individuals' difference scores is assumed to be a normal distribution)

The significance level cutoff from the t table is not accurate.

With a t test for dependent means...

What we call Population 2 will ordinary have a mean of 0 (the mean of the population of the difference scores is 0) Comparing the population of difference scores that your sample of difference scores comes from to a population of difference scores with a mean of 0.

How can you tell if you have violated the normal curve assumption for a t test for dependent means?

You look at the distribution of the sample of difference scores to see if it is dramatically different from a normal curve.

SEE TABLE 9 comparing

Z-tests, to t Tests for a Single Sample, to t Tests for Dependent Means

Repeated measures design

research strategy in which each person is tested more than once; same as within-subjects design common example: when you measure the same people before and after some psychological or social intervention (experimental intervention)