Notes

Outline

Statistics 7 t-Tests Sherril M. Stone, Ph.D. Department of Family Medicine OSU-College of Osteopathic Medicine

t-Test t-Test – hypothesis testing technique to reach conclusion about m based on sample results z uses σ (pop SE) t uses s (sample SE)

t Distribution for Hypothesis Testing Table values – mathematically derived theoretical distribution values Significance level – setting the a a is probability (p) value – (.05, .01, .001) - .05 = 5 (out of 100) .p < a  -  reject H0, accept H1 (this is statistically significant) .p > a  -  accept H0, reject H1   (this is NOT statistically sig) Rejection Region reject H0 if X  significantly different from m (IV had affect) If df not in table – use next lower t .05 (14 df)obs = 3.862 ≥ t .05 (14 df)tab = 2.145 (reject H0) t .05 (14 df)obs = 1.789 ≤ t .05 (14 df)tab = 2.145 (accept H0)

One Sample t-test t   =    X- m0       SE Interpretation – always uses terms of experiment and direction of difference Generalizability – is it random sample & does it apply to similar samples? Uses of t Checking claims, norms, no-error standards, etc Simplest test for hypothesis testing

Examples for One-Sample t-test EXAMPLE 1 N = 16 __  X   =  181.62 m0     =  180.19 SD  = 1.6 SE  =    1.6              √N t   =

EXAMPLE 3 - Calculate t for the following data. (you will have to calculate all needed stats) μ = 4.6 4 6 8 9 5 3 7 5 2 7 6 1 6 6 9 4 7 7 2 8 6 8 3 5 7 4 7 6 4 7

EXAMPLE 4 - Calculate t for the following data. (you will have to calculate all needed stats) μ = 14.8 10 34 16 19 16 17 17 25 28 17 35 33 26 26 32 14 16 21 22 27 19 28 28 27 27 33 26 27 24 15

Effect Size for t statistic

Examples of Effect Size Example 1 N  = 38 df = 38-1 = 37 t  = 5.3 Calculate r2

Experimental Methods for Two-Sample t-tests Experimental method – mean of one group is compared to mean of a different group after some type of treatment has been applied to either group. Independent variable – manipulated variable Dependent variable – measured variable Treatment – the manipulation Experimental (treatment) group – the group receiving the treatment Control group – the group not receiving treatment

Hypothesis Testing for 2-Sample t-test Experiment Define H0 and H1 H0:  X  =  μ H1:  X  ¹  μ Assume treatment has no effect (H0) Choose a (.05 Behavioral/.01 or .001 Medical Sciences) Choose correct inferential statistic Calculate the statistics for your sample data Compare calculated results to the Table value Write interpretation using terms of experiment

Types of 2-Sample Designs You must know the design before you analyze data IV and DV will not indicate the design Different t-tests used for each design Independent Samples – no reason to pair the data Correlated Samples – matches the data Natural pairs Matched pairs (split-litter for animals, twins, siblings) Repeated measure If t-test is significant – then difference in participant is due to the treatment

Independent Samples There no reason to pair the data Randomly assign participants to groups One group receives treatment, other group does not If t-test is significant – treatment had an effect

2-Sample Formulas

Example 1 - Independent Samples You are interested in the effects of an anti-anxiety drug on weight loss. The drug group (7 rhesus monkeys) received the drug while the control (placebo) group (6 rhesus monkeys) was given a sugar pill (placebo). The monkeys were tested every Monday for 7 weeks. Their results served as the dependent variable. The H0 was that the drug had no effect on weight. The H1 was that the drug had an effect on weight.

Correlated Samples Natural pairs researcher does not pair naturally occurring pair pairing is based on logical, memberships, etc Matched pairs (split-litter for animals, twins, siblings) researcher controls the pairing may pretest to determine pair mate randomly assign each pair mate to treatment & control Repeated measure pre and post test design each participant serves as own control If t-tests are significant – then difference in participants is due to the treatment

Correlated Samples Formulas

Example 2 – Repeated Measures You are interested in the effects of a new diabetes drug for pre-teen  girls. You take a sample of blood from a group of 14 Caucasian 12-year old girls then give them a dosage of the drug. The girls spend the day together and eat the same meals. They monitor their insulin levels during the day as usual. At the end of the day, you take another sample of blood. The data consists of 14 pairs of pre and post insulin levels. The H0 is that the new drug did not affect the insulin levels. The H1 is that after the new drug, the girls’ insulin level will decrease.

Example 3 – Natural Pairs You are interested in the number of viral infections that the average married couple experiences in a year. You ask 30 couples to participate in your study. You separate the spouses and ask them how many colds they have in the past year. Your H0 was that the gender does not make a difference in the number of colds reported. The H1 was that the females experience more colds than the males.

Effect Size for 2-Samples

Example 4 – Independent Samples Stretching before running may decrease quad strains thus reducing the need for medication. Athletes in the same physical condition and with the same running time volunteered to participate in the study. Group 1 did not stretch but Group 2 stretched. Both groups ran 3 miles everyday for 2 weeks (14 days). The number of muscle strains for each runner is presented. Determine if the groups significantly differ – i.e. if stretching was effective in decreasing quad strains and medication.

Example 5 – Independent Samples The researcher in Example 4 decided to replicate her results by conducting a second study. However, before the run, 3 runners dropped out of the control group and 2 joined the experimental group.  The researcher decided to conduct her study anyway. Determine if the groups significantly differ – i.e. was stretching effective and reduced quad strains and medication.

Example 6 – Repeated Measures Sample Fatty food is hypothesized to correlate with obesity. Your obese patients (all 75 lbs. overweight) are weighed before beginning your study (pre-weight) and again after 1 year of not eating any fatty food on a list you provide for them (all other lifestyle behaviors remain the same). Your patients again are weighed at the end of the year. Their pre- and post-weights are presented. Determine if the pre weight significantly correlated to the post weight – i.e. does consumption of fatty foods predict obesity.

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