invesitagat - Chapter 5 Investigating the Difference in...

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Unformatted text preview: Chapter 5: Investigating the Difference in Scores Outline: • Introduction • Why Use Tests of Difference • Dependent and Independent Variables • Null Hypothesis – One & Two Tailed Tests of Significance • Types of Errors – Type I and Type II Error • • • • Standard Error of Mean Standard Error of the Difference of the Means Assumptions When Testing for Difference Types of t-Tests – Independent & Dependent Groups • Analysis of Variance – One Way ANOVA, Repeated Measures ANOVA, Post Hoc Tests • Selecting the Test Objectives: 1. Explain how you could use tests of difference. 2. Define Type I and Type II errors and how it relates to level of significance. 3. List the assumptions that are required to test for differences between means. 4. Understand when to use and how interpret the t-test for independent and dependent groups. 5. Understand when to use and how to interpret analysis of variance for independent groups and analysis of variance for repeated measures. Introduction • Questions? • Is there a difference between groups? – males vs. females – athletes vs. non-athletes – treatment method of an injury ice vs. ultrasound – flexibility programs static stretching vs. propriomuscular stretching – interval vs. continuous training programs Why use tests of difference? 1. Improve your understanding and interpretation of research. 2. Allows for the evaluation of the effects of a cause or treatment. 3. In experimental research – allows for development of cause-and-effect relationships. Dependent and Independent Variables • Dependent variable: • What you are measuring to determine if it changes. • Independent variable: • What you are doing, controlling or manipulating that might cause change to the dependent variable. Check your Understanding • Identify the dependent and independent variables: – The effect of background music on endurance performance during cycle ergometry to fatigue. – Direct and indirect coaching styles on high school soccer players’ decision making skills during competition. – The effect of ankle bracing on peak mediolateral ground reaction force during cutting maneuvers in collegiate male basketball players. – Arm versus combined leg & arm exercise: Blood pressure responses and ratings of perceived exertion at the same indirectly determined heart rate. The Null Hypothesis • • • • • Hypothesis: A prediction about what will happen. A scientific hunch - expected outcome. Null hypothesis: no difference between groups (ie., means are equal ; Ho: X1 = X2). • Alternative hypothesis: • there is a difference between groups (means are equal ; Ha: X1 ≠ X2). • Directional Hypothesis – the difference is expected to occur in a certain direction (X1 > X2 or X1 < X2). minutes of exercise and weight loss, high intensity vs. low intensity One & Two Tailed Levels of Significance • Tables will list: • t value for accepting or rejecting the hypothesis (usually pre-selected at the 0.05 or 0.01 level). • t value – when the calculated value is > the table value, we can say “ ”. the difference between the means is significant” - “it is real difference”. • Degrees of Freedom (df) - determined by the sample size (N – 1; where N = sample size). • When using a t-test (with two independent groups) • df = (N1 - 1) + (N2 - 1) or N1 + N2 - 2 This Table – Berg & Latin (1994). Table D – page 226. Critical Values of t. Text – Appendix B – page 276. Critical Values of t (Two-Tailed Large samples df will be Test). higher, t-values slightly smaller. Less of a difference between means will be statistically significant. Illustration of 0.05 Level of Significance Is a difference (p < 0.05) Accept that there is no difference (p > 0.05) 0 % Ho: X1 ≠ X2 Ho: X1 = X2 100 % no difference in means • Significant – differences between the scores are real at the alpha level identified. – 5 chances in 100 that you are in error. • Nonsignificant – there is no real difference between the scores. One-tailed or Two-tailed - Which Version Should You Use? • One-tailed - when you believe the mean difference will occur in one direction. – Strength – Training - pre and post training • Two-tailed – when the difference can be in either direction. – Strength – comparing two groups with different strength programs (not sure which is a better program) Take many samples from a population, means of the samples would create a normal distribution. Do Not Reject Null Hypothesis Do Not Reject Null Hypothesis Reject Null Hypothesis One Tailed Reject Null Hypothesis Reject Null Hypothesis Two Tailed Types of Errors • Type I Error – • Reject a null hypothesis (X1 = X2) when it is true. • Conclude that there is a difference when there really isn’t a difference. • Occurs more often when significance = 0.05 • Type II Error – • Accept the null hypothesis when it is false. • Conclude that there was no difference when there was a real difference. • Occurs more often when significance = 0.01 Level of Significance and Type of Error True state in population Your decision Null Hypothesis (H0) is true Alternate hypothesis (H1) is false Null Hypothesis (H0) is false Alternate hypothesis (H1) is true Reject H0 Accept H1 Type I Error (alpha) Correct decision Accept H0 Reject H1 Correct decision Type II Error (beta) • Can reduce your chance of making a Type I error by increasing the level of significance - 0.01, 0.001. • Best way of reducing a Type II error is to increase the sample size. Standard Error of Mean (S.E.M.) • Represents the standard deviation of the sample distribution. • Formula S.E.M. = s N • Where: s = standard deviation of sample; N = number of scores. • ± 1 SEM = range interpreted as the limits of the 68% confidence intervals for mean. Mean = 10m SEM = 2. 68% of the time we would find a mean between 8-12 Standard Error of the Difference between Means • • Represents the standard deviation of all the observed differences between pairs of sample means. Estimate of the expected difference between two sample means randomly drawn from the same population. • Formula Sx1-x2 = SEM12 + SEM22 • Where: Sx1-x2 = standard difference of the means; SEM1 of sample 1, SEM2 of sample 2 Assumptions When Testing for Differences (t-Test & ANOVA) 1. Data are drawn from normally distributed population 2. Data represent random samples from populations. 3. Variance in each group is similar. variance = difference between the means Bigger difference between means = larger t ratio t ratio = Variance between Groups Variance within Groups spread of scores within the group Bigger variance (spread of scores) = smaller t ratio Types of t Tests 1. Independent t Test • Do two sample (group) means differ from each other? Two groups – each assessed on Leger m Shuttle Run (“Beep” test). The • Most often used t test. 20 predicted VO max in ml/kg/min are 2 given below. Group 1 42 50 57 45 56 69 45 43 46 51 61 55 40 Group 2 47 51 59 43 47 43 46 37 30 63 48 53 53 44 40 44 40 38 64 44 48 37 41 51 30 44 39 34 48 50 30 37 39 47 55 49 43 Group 1 Group 2 Mean 48.92 Mean 43.72 Low 30 Low 30 High 69 High 63 SD 8.78 SD 7.83 Range 39 Range 33 Calculation Steps: Independent t Test 1. Calculate descriptive statistics. • Group 1: Mean = 48.92, SD = 8.78, N = 25 • Group 2: Mean = 43.72, SD = 7.83, N = 25 2. Calculate the SEM for each group. • SEM1 = s1 = 8.78 = 1.756 N1 25 • SEM2 = s2 = 7.83 = 1.566 N1 25 3.Calculate the standard error of difference between groups. • sx-x = SEM12 + SEM22 = (1.756)2 + (1.566)2 = 2.353 4.Calculate the t-ratio by substituting the values in the formula • t = X1 - X2 = 48.92 – 43.72 = 2.213 sx-x 2.353 t – compare to Critical Value of t in Table Degrees of freedom = (N1 - 1) + (N2 - 1) or = N1 + N 2 - 2 df = (25 – 1) + (25 – 1) = 48 t = 2.213 Two Tailed 2.213 > 2.060 so this difference is significant at the 0.05 level. The means are different. 1. Independent t Test - Excel T test 2 Sample Assuming Equal Variance Group 1 Group 2 Group 2 42 50 57 45 56 69 45 43 46 51 61 55 40 47 59 47 46 30 48 53 40 40 64 48 41 Group 1 51 43 43 37 63 53 44 44 38 44 37 51 30 44 39 34 48 50 30 37 39 47 55 49 43 Mean 48.92 Mean 43.72 Low 30 Low 30 High 69 High 63 SD 8.78 SD 7.83 Range 39 Range 33 Degrees of freedom = (N1 - 1) + (N2 - 1) or = N 1 + N2 - 2 probability of making a mistake - 3 % chance of being an error t-Test: Two-Sample Assuming Equal Variances Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Group 1 48.92 77.16 25 69.26833333 0 48 2.208975917 0.015991463 1.677224197 0.031982925 2.010634722 Group 2 43.72 61.37666667 25 SPSS Output Group Statistics Group N VO2max 1 25 2 25 Mean Std. Deviation 48.92 8.784 43.72 Std. Error Mean 1.757 7.834 1.567 Independent Samples Test Levene's Test for Equality of Variances VO2max Equal variances assumed Equal variances not assumed F .381 Sig. .540 t-test for Equality of Means t 2.209 2.209 df 48 47.385 95% Confidence Interval of the Sig. (2Mean Std. Error Difference tailed) Difference Difference Lower Upper .032 5.200 2.354 .467 9.933 .032 5.200 2.354 .465 9.935 ex. North american football team -comparing two individuals with similar strength 2. Dependent t Test • Do the scores of two sets of data, that are , related in some way differ from each other? • Relationship takes one of two forms: a) Two groups are matched on one or more characterist and are thus not independent. b) One group is tested twice on the same variable ( pretest and postest ). • is used more often with dependent t-tests – scores are expected to One-tailed test increase or decrease. Dependent t Test Formula • Formula is: • t= ( D ÷ N) 1 ÷ N [N( D2] – ( D)2 (N – 1) • Where: D = difference in paired scores (ie., pretest – post test), N = number of paired scores • Degrees of freedom (df) = N – 1 Example: Dependent t Test Free Throw scores on 25 attempts effect of 4 weeks free throw practice 2 Subject Pretest Posttest D D 1 19 20 -1 1 2 17 15 2 4 3 19 20 -1 1 4 14 16 -2 4 5 13 17 -4 16 6 16 16 0 0 7 16 15 1 1 8 17 18 -1 1 9 17 19 -2 4 10 14 17 -3 9 ∑ 162 173 -11 41 Example: Dependent t Test • t= (-11 ÷ 10) 1 ÷ 10 [10(41] – (-11)2 (10 – 1) • t= -1.10 = -1.1 0.10 410 – 121 0.10 289 9 9 = -1.1 -1.1 = -1.1 = -1.941 • 0.10 32.11 0.10 x 5.667 0.5667 • Compare -1.94 to One tailed level of significance 0.05 level column @ df (10 – 1) = 9. • 1.94 > 1.833 so the difference between (The means are different) the pretest and posttest mean is significant @ 0.05 level. • 4 wks practice improved free throw shooting. 3. Dependent t Test - Excel Pre Test 51 43 43 37 63 53 44 44 38 44 37 51 30 44 39 34 48 50 30 37 39 47 55 49 43 Post Test 42 50 57 45 56 69 45 43 46 51 61 55 40 47 59 47 46 30 48 53 40 40 64 48 41 t-Test: Paired Two Sample for Means Pre Test Mean Variance Observations Pearson Correlation Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Degrees of freedom = (N – 1) = 25 – 1 = 24 43.72 61.37666667 25 0.242453497 0 24 -2.535328822 0.009081853 1.710882067 0.018163705 2.063898547 Post Test 48.92 77.16 25 Analysis of Variance (One-way ANOVA) • Allows for evaluation of 2 or more group means on one dependent variable (an extension of the independent t-test ). • F-ratio - the calculated number associated with ANOVA ( like critical t value for t-test ). • Null Hypothesis: M1 = M2 = M3 Bottaro, M., Martins, B., Gentil, P., and Wagner, D. (2009). Effects of rest duration between sets of resistance training on acute hormonal responses in trained women. Journal of Sci. and Med. in Sport, 12: 73-78. • Methods: • Standard statistical procedures were used to calculate means and standard deviations (S.D.). Differences in hormonal responses among time point for each trial were evaluated using a one-way ANOVA with repeated measures. The resulting integrated area under the response curve for GH (GHauc) and cortisol (Cauc) were computed using a trapezoidal method after pre-exercise values were subtracted from each time point. Differences among GHauc and among Cauc rest intervals (30, 60, and 120 s) were analyzed using a one-way ANOVA with repeated measures. Multiple comparisons with confidence interval adjustment by the LSD (Least Significant Difference) method were used as post hoc when necessary. The significance level was set at p < 0.05. The SPSS 14.0 (SPSS, Chicago, IL) was used in the current analyses. ANOVA Table • SS = sum of squares • Calculations – each subject’s score is squared. Sum of squared scores is used in the calculations. • Treatment variance (Between group variance) • Error variance (Within group variance) • Total variance • MS = mean squares: SS divided by the appropriate df to calculate MS Degrees of Freedom • Between Groups (Treatment) df = k – 1 where: k = number of groups • Within Groups (Error) df = N – k where: N = total number of subjects in the study and k = number of groups • Total df = N – 1 where: N = total number of subjects variance due to the treatment in the study • F = Between Group (Treatment) Variance • Within Group (Error) Variance variance due to chance or sampling error Analysis of Variance Group 1 Group 2 Group 3 12 7 13 15 10 14 10 11 10 11 8 9 9 9 12 14 10 11 12 12 11 13 9 15 Anova: Single Factor SUMMARY Groups Group 1 Group 2 Group 3 ANOVA Source of Variation Between Groups Within Groups Total Count 8 8 8 SS 31.75 74.875 106.625 Sum Average Variance 96 12 4 76 9.5 2.571429 95 11.875 4.125 df MS F P-value F crit 15.875 4.452421 0.024435 3.4668 2 21 3.565476 23 3 Classes – knowledge on personal health (quiz/15). There is a difference in scores (p < 0.05) between the groups 1, 2, and 3. SPSS Outputs Descriptives Score N 1 2 3 Total 8 8 8 24 Mean 12.00 9.50 11.88 11.13 Std. Deviatio Std. Error n 2.000 .707 1.604 .567 2.031 .718 2.153 .440 95% Confidence Interval for Mean Lower Upper Minimu Maximu Bound Bound m m 10.33 13.67 9 15 8.16 10.84 7 12 10.18 13.57 9 15 10.22 12.03 7 15 ANOVA Score Between Groups Within Groups Total Sum of Squares 31.750 2 Mean Square 15.875 74.875 21 3.565 106.625 23 df F 4.452 Sig. .024 Group 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 Score 12 15 10 11 9 14 12 13 7 10 11 8 9 10 12 9 13 14 10 9 12 11 11 15 Post-Hoc Tests • Significant F-ratio = means are significantly different but does not indicate which group is different. • A post-hoc test is performed to determine which group is different from the others. Groups Count Sum Average Variance Group 1 8 96 12.0 4.00 Group 2 8 76 9.50 2.57 Group 3 8 95 11.88 4.13 • Perform similar functions as an independent t-test. • Tests are listed from liberal to stringent. Names of Post-Hoc Tests: 1. Duncan Multiple Range difference between means 2. Newman-Keuls 3. Fisher's Least Significant Difference (LSD) 4. Tukey's Honestly Significant Difference (HSD) 5. Scheffe' minimize the chance of making a Type I error SPSS Output Post Hoc Tests Multiple Comparisons Score Tukey HSD (I) Group (J) Group 1 dimension 2 3 3 2 dimension 1 dimension2 3 3 3 dimension 1 3 2 *. The mean difference is significant at the 0.05 level. group 1 & 2 sig. difference group 1 & 3 Not group 2 & 3 just not Mean Difference (IStd. Error J) * 2.500 .944 .125 .944 * -2.500 .944 -2.375 .944 -.125 .944 2.375 .944 95% Confidence Interval Sig. .038 .990 .038 .051 .990 .051 Lower Bound Upper Bound .12 4.88 -2.25 2.50 -4.88 -.12 -4.75 .00 -2.50 2.25 .00 4.75 group 1 Multiple Comparisons Score Scheffe (I) Group (J) Group Mean Difference (I-J) Std. Error 1 2.500* .944 dimens 2 .125 .944 ion3 3 2 -2.500* .944 dimens 1 dimension2 -2.375 .944 ion3 3 3 -.125 .944 dimens 1 2.375 .944 ion3 2 *. The mean difference is significant at the 0.05 level. 95% Confidence Interval Sig. .049 .991 .049 .063 .991 .063 Lower Bound Upper Bound .01 4.99 -2.36 2.61 -4.99 -.01 -4.86 .11 -2.61 2.36 -.11 4.86 Repeated Measures ANOVA • The same subjects are tested several times (dependent variable) to determine the effect of the independent variable. • For example - measure changes in a variable with time ( pre, mid, post season). • When two scores are assessed = dependent t-test for two means. Selecting the Statistical Test 2 Groups 1 Group 2 or More Groups are matched Independent Independent t Test Dependent Paired t Test 2 Tests Dependent t Test or Paired Samples Modified from Figure 14.3, Baumgartner (2006). 1–Way ANOVA > 2 Groups - Post Hoc Tests > 2 Tests - ANOVA Repeated Measures & Post Hoc Tests ...
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