Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
t-Tests Interval Estimation and the t Distribution Large Sample z-Test • Sometimes we have reason to test hypotheses involving specific values for the mean. – Example 1. Claim: On average, people sleep less than the often recommended eight hours per night. – Example 2. Claim: On average, people drink more than the recommended 2 drinks per day. – Example 3. Claim: On average, women take more than 4 hours to run the marathon. • However, it is rare that we have a specific hypothesis about the standard deviation of the population under study. • For these situations, we can use the sample standard deviation s as an estimator for the population standard deviation s. • If the sample size is pretty big (e.g., >100), then this estimate is pretty good, and we can just use the standard z test. PSYC 6130, PROF. J. ELDER 3 Example: Canadian General Social Survey, Cycle 6 (1991) PSYC 6130, PROF. J. ELDER 4 But what if we don’t have such a large sample? PSYC 6130, PROF. J. ELDER 5 Student’s t Distribution • Problem: for small n, s is not a very accurate estimator of s. • The result is that the computed z-score will not follow a standard normal distribution. • Instead, the standardized score will follow what has become known as the Student’s t distribution. X - t sX PSYC 6130, PROF. J. ELDER 6 where s X s n Student’s t Distribution Normal distribution t distribution, n=2, df=1 t distribution, n=10, df=9 t distribution, n=30, df=29 How would you describe the difference between the normal and t distributions? PSYC 6130, PROF. J. ELDER 7 Student’s t distribution • Student’s t distribution is leptokurtic – More peaked – Fatter tails • What would happen if we were to ignore this difference, and use the standard normal table for small samples? PSYC 6130, PROF. J. ELDER 8 Student’s t Distribution • Critical t values decrease as df increases • As df infinity, critical t values critical z values • Using the standard normal table for small samples would result in an inflated rate of Type I errors. PSYC 6130, PROF. J. ELDER 9 One-Sample t Test: Example PSYC 6130, PROF. J. ELDER 10 PSYC 6130, PROF. J. ELDER 11 Reporting Results • Respondents who report being very forgetful sleep, on average, 7.11 hours/night, significantly less than the recommended 8 hours/night, t(37)=2.25, p<.05, twotailed. PSYC 6130, PROF. J. ELDER 12 Confidence Intervals • NHT allows us to test specific hypotheses about the mean. – e.g., is < 8 hours? • Sometimes it is just as valuable, or more valuable, to know the range of plausible values. • This range of plausible values is called a confidence interval. PSYC 6130, PROF. J. ELDER 13 Confidence Intervals • The confidence interval (CI) of the mean is the interval of values, centred on the sample mean, that contains the population mean with specified probability. • e.g., there is a 95% chance that the 95% confidence interval contains the population mean. • NB: This assumes a flat prior on the population mean (nonBayesian). X Confidence Interval PSYC 6130, PROF. J. ELDER 14 Confidence Intervals p( ) sX p .025 PSYC 6130, PROF. J. ELDER X 95% Confidence Interval 15 p .025 Basic Procedure for Confidence Interval Estimation 1. Select the sample size (e.g., n = 38) 2. Select the level of confidence (e.g., 95%) 3. Select the sample and collect the data (Random sampling!) 4. Calculate the limits of the interval t X X sX t sX X s X t / 2 X s X t / 2 PSYC 6130, PROF. J. ELDER 16 End of Lecture 4 Oct 8, 2008 Selecting Sample Size • Suppose that 1. You have a rough estimate s of the standard deviation of the population, and 2. You want to do an experiment to estimate the mean within some 95% confidence interval of size W. Then the sample size n should be roughly 4s n W 2 PSYC 6130, PROF. J. ELDER 18 Assumptions Underlying Use of the t Distribution for NHT and Interval Estimation • Same as for z test: – Random sampling – Variable is normal • CLT: Deviations from normality ok as long as sample is large. – Dispersion of sampled population is the same as for the comparison population PSYC 6130, PROF. J. ELDER 19 Sampling Distribution of the Variance Sampling Distribution of the Variance • We are sometimes interested in testing a hypothesis about the variance of a population. – e.g., is IQ more diverse in university students than in the general population? Suppose we measure the IQ of a random sample of 13 university students We then calculate the sample variance: ( X X ) 2 s 2 N 1 400 Suppose that we know the variance s 02 of IQs in the general population: s 02 152 225 Can we conclude that student IQs are more diverse? To solve this problem we need to know the range of plausible values for the test statistic s 2 under the null hypothesis. PSYC 6130, PROF. J. ELDER 21 Sampling Distribution of the Variance • What form does the sampling distribution of the variance assume? p(s2) • If the variable of interest (e.g., IQ) is normal, the sampling distribution of the variance takes the shape of a c-squared distribution: 0 PSYC 6130, PROF. J. ELDER E (s 2 ) s 2 s2 22 Sample Variances and the c-Square Distribution We first standardize the sample variance statistic by multiplying by the degrees of freedom and dividing by the population variance s 2 : s 2 (n 1)s 2 c s2 s2 2 The resulting variable c 2 follows a c 2 ( ) distribution with df n 1. p(c2) =9 =29 =99 0 PSYC 6130, PROF. J. ELDER 50 100 c2 23 150 Sample Variances and the c-Square Distribution • The c-square distribution is: – strictly positive. – positively skewed. • Since the sample variance is an unbiased estimator of the population variance: E(s2) = s 2 • Due to the positive skew, the mean of the distribution E(s2) is greater than the mode. • As the sample size increases, the distribution approaches a normal distribution. • If the original distribution is not normal and the sample size is not large, the sampling distribution of the variance may be far from csquare, and tests based on this assumption may be flawed. PSYC 6130, PROF. J. ELDER 24 Example: Height of Female Psychology Graduate Students Canadian Adult Female Population: s 2005 PSYC 6130A Students (Female) 63.937 in 2.7165 in Canadian Adult Male Population: s 69.252 in 3.189 in n 131,110! Source: Canadian Community Health Survey Cycle 3.1 (2005) Caution: self report! PSYC 6130, PROF. J. ELDER 25 Properties of Estimators • We have now met two statistical estimators: X is an estimator for . s 2 is an estimator for s 2 . Both of these estimators are: Unbiased, i.e., E( X )= E (s 2 )= s 2 Consistent, i.e., the quality of the estimate improves as the sample size increases. Efficient, i.e., given a fixed sample size, the accuracy of these estimators is better than competing estimators. PSYC 6130, PROF. J. ELDER 26 NHT for Two Independent Sample Means Conditions of Applicability • Comparing two samples (treated differently) • Don’t know means of either population • Don’t know variances of either population • Samples are independent of each other PSYC 6130, PROF. J. ELDER 28 Example: Height of Canadian Males by Income Category (Canadian Community Health Survey, 2004) PSYC 6130, PROF. J. ELDER 29 Sampling Distribution To solve this problem we need to know the sampling distribution for the difference of the means, i.e., X1 X 2 Under the null hypothesis, both samples come from the same distribution. Suppose this distribution is normal. Then we know that X1 and X 2 are also normally distributed: X1 is N( ,s X1 ) N( , X 2 is N( ,s X2 ) N( , PSYC 6130, PROF. J. ELDER s1 n1 ) s2 n2 ) 30 Sampling Distribution (cntd…) Major Theorem of Probability: Any linear combination of normal variables is itself normal. Thus X1 X 2 is also normal: X1 X 2 is N (0,s X1 X2 ) What is the dispersion s X1 X2 ? Basic principle for normal distributions - variances add: s X2 X s X2 s X2 1 2 1 2 Knowing the standard error for the 2 distributions, we can calculate our sampling distribution. z X 1 X 2 1 2 s X X 1 2 PSYC 6130, PROF. J. ELDER 31 NHT for Two Large Samples Recall: If sample is large (e.g., n 100), can approximate population variance by sample variance: s X2 s X2 1 s 2 X2 1 s 2 X2 And thus we can estimate s X21 X2 s X2 1 s X2 2 PSYC 6130, PROF. J. ELDER 32 Height of Canadian Males by Income Category (Canadian Community Health Survey, 2004) X 69.87 " s 2.63 " n 7586 X 69.01" s 2.85" n 7777 PSYC 6130, PROF. J. ELDER 33 NHT for Two Small Samples Example: Social Factors in Psychological Well-Being Canadian Community Health Survey, 2004 PSYC 6130, PROF. J. ELDER 35 Social Factors in Psychological Well-Being (cntd…) Canadian Community Health Survey, 2004 PSYC 6130, PROF. J. ELDER 36 Social Factors in Psychological Well-Being (cntd…) Canadian Community Health Survey, 2004: Respondents who report never getting along with others PSYC 6130, PROF. J. ELDER 37 NHT for Two Small Independent Samples By analogy with one-sample NHT, we might approximate the standard errors s X1 and s X2 by the sample standard errors s X1 and s X2 . Unfortunately, the resulting sampling distribution of the difference of the means is not straightforward to analyze. So what do we do? If we can assume homogeneity of variance (the two populations have the same variance), then there is a statistic that follows the t distribution and is simple to analyze. PSYC 6130, PROF. J. ELDER 38 NHT for Two Small Independent Samples (cntd…) If both populations have the same variance, we want to use both samples simultaneously to get the best possible estimate of this variance. In general, recall that s 2 SS n 1 2 2 n 1 s n 1 s SS SS 1 1 2 2 1 2 Thus our formula for the pooled variance is sp2 n1 1 n2 1 n1 1 n2 1 And the sample standard error is s X and t 1 X 2 1 2 s X1 X 2 PSYC 6130, PROF. J. ELDER 2 X1 X 2 sp2 n1 sp2 n2 follows a t distribution with n1 n2 2 degrees of freedom. 39 Pooled Variance Pooled variance is s 2 p n1 1 s12 n2 1 s22 n1 1 n2 1 df1s12 df2s22 df1 df2 Note that the pooled variance is a weighted sum of the sample variances. The weights are proportional to the size of each sample (Bigger samples are more reliable estimators of the common variance) df2 df1 2 1 s s PSYC 6130, PROF. J. ELDER 40 2 p s 22 Social Factors in Psychological Well-Being (cntd…) Canadian Community Health Survey, 2004: Respondents who report never getting along with others X 36.59 s 22.40 n 37 X 41.84 s 23.87 n 25 PSYC 6130, PROF. J. ELDER 41 Reporting the Result No significant difference was found between the psychological well-being of men (M 41.8, SD 23.9) and women ( M 36.6, SD 22.4) who report never getting along with others, t (60) 0.88, p .38. PSYC 6130, PROF. J. ELDER 42 Confidence Intervals for the Difference Between Two Means X t 1 X 2 1 2 s X1 X 2 1 2 X1 X 2 ts X1 X 2 1 2 X1 X 2 tcrit s X1 X 2 p( 1 2 ) s X1 X2 p .025 X1 X 2 t.025 PSYC 6130, PROF. J. ELDER p .025 95% Confidence Interval t.025 43 Underlying Assumptions • Dependent variable measured on interval or ratio scale. • Independent random sampling – (independence within and between samples) – In experimental work, often make do with random assignment. • Normal distributions – Moderate deviations ok due to CLT. • Homogeneity of Variance – Only critical when sample sizes are small and different. PSYC 6130, PROF. J. ELDER 44 End of Lecture 5 Oct 15, 2008 Social Factors in Psychological Well-Being (cntd…) Canadian Community Health Survey, 2004: Respondents who report never getting along with others X 36.59 s 22.40 n 37 X 41.84 s 23.87 n 25 PSYC 6130, PROF. J. ELDER 46 Separate Variances t Test • If – Population variances are different (suggested by substantially different sample variances) AND – Samples are small AND – Sample sizes are substantially different • Then – Pooled variance t statistic will not be correct. • In this case, use separate variances t test PSYC 6130, PROF. J. ELDER 47 Separate Variances t Test X t 1 X 2 1 2 s X1 X 2 where s X2 1 X2 s X2 1 s X2 2 • This statistic is well-approximated by a t distribution. • Unfortunately, calculating the appropriate df is difficult. • SPSS will calculate the Welch-Satterthwaite approximation for df as part of a 2-sample t test: s df 2 X1 s X4 1 df1 s 2 X2 2 s X4 2 df2 PSYC 6130, PROF. J. ELDER 48 Social Factors in Psychological Well-Being (cntd…) Canadian Community Health Survey, 2004: Respondents who report never getting along with others X 36.59 s 22.40 n 37 X 41.84 s 23.87 n 25 PSYC 6130, PROF. J. ELDER 49 Summary: t-Tests for 2 Independent Sample Means n1, n2 100 n1 s2 Test n2 s1 statistic s X2 1 X 2 df t s12 s22 n1 n2 WelchSatterthwaite t s p2 n1 n2 2 n1 s p2 n2 t s12 s22 n1 n2 n1 n2 2 t s12 s22 n1 n2 n1 n2 2 z s12 s22 n1 n2 NA z s12 s22 n1 n2 NA z s12 s22 n1 n2 NA z s12 s22 n1 n2 NA PSYC 6130, PROF. J. ELDER 50 More on Homogeneity of Variance • How do we decide if two sample variances are different enough to suggest different population variances? • Need NHT for homogeneity of variance. – F-test • Straightforward • Sensitive to deviations from normality – Levene’s test • More robust to deviations from normality • Computed by SPSS PSYC 6130, PROF. J. ELDER 51 Levene’s Test: Basic Idea 1. Replace each score X1i , X 2i with its absolute deviation from the sample mean: d1i | X1i X1 | d 2 i | X 2 i X 2 | 2. Now run an independent samples t-test on d1i and d2i : t d1 d 2 sd1 d2 SPSS reports an F-statistic for Levene’s test • Allows the homogeneity of variance for two or more variables to be tested. • We will introduce the F distribution later in the term. PSYC 6130, PROF. J. ELDER 52 The Matched t Test Independent or Matched? • Application of the Independent-Groups t test depended on independence both within and between groups. • There are many cases where it is wise, convenient or necessary to use a matched design, in which there is a 1:1 correspondence between scores in the two samples. • In this case, you cannot assume independence between samples! • Examples: – Repeated-subject designs (same subjects in both samples). – Matched-pairs designs (attempt to match possibly important attributes of subjects in two samples) PSYC 6130, PROF. J. ELDER 54 Example: Assignment Marks A3 A4 72 80 70 80 69 93 83 88 88 93 88 93 87 88 88 93 85 100 85 100 70 80 72 90 60 80 83 75 81 83 68 75 36 93 80 100 65 83 65 83 41 75 73 88 68 75 Assignment 2 Mark (%) 100 90 80 70 60 50 40 40 60 Assignment 1 Mark (%) Mean 73 86 SD n 80 14 8 23 23 These scores are not independent! PSYC 6130, PROF. J. ELDER 55 100 Better alternative: The matched t-test using the direct difference method A3 A4 A4-A3 72 80 70 80 69 93 83 88 88 93 88 93 87 88 88 93 85 100 85 100 70 80 72 90 60 80 83 75 81 83 68 75 36 93 80 100 65 83 65 83 41 75 73 88 68 75 Mean SD n 8 10 24 5 5 5 1 5 15 15 10 18 20 -8 2 7 57 20 18 18 34 15 7 73 86 13 14 23 8 23 13 23 PSYC 6130, PROF. J. ELDER t X 0 s/ n t 56 D 0 sD / n Matched vs Independent t-test • Why does a matched t-test yield a higher t-score than an independent t-test in this example? – The t-score is determined by the ratio of the difference between the groups and the variance within the groups. – The matched t-test factors out the portion of the within-group variance due to differences between individuals. PSYC 6130, PROF. J. ELDER 57 The Matched t Test and Linear Correlation • The degree to which the matched t value exceeds the independentgroups t value depends on how highly correlated the two samples are. • Alternate formula for matched standard error: 100 Assignment 4 Mark (%) s12 s22 2rs1s2 s , n n where r is the Pearson correlation 2 D 90 80 r2 = 0.18 70 60 50 40 40 60 80 Assignment 3 Mark (%) PSYC 6130, PROF. J. ELDER 58 100 Case 1: r = 0 • Independent t-test s 2 X1 X 2 1 s12 s22 n • Matched t-test s12 s22 2rs1s2 s n n 1 2 s1 s22 n 2 D Thus the t-score will be the same. But note that df 2(n 1) df n 1 Thus the critical t-values will be larger for the matched test. PSYC 6130, PROF. J. ELDER 59 Case 2: r > 0 • Independent t-test s 2 X1 X 2 1 s12 s22 n • Matched t-test s12 s22 2rs1s2 s n n 2 D Now the t-score will be larger for the matched test. Although the critical t-values are larger, the net result is that the matched test will often be more powerful. PSYC 6130, PROF. J. ELDER 60 Confidence Intervals • Just as for one-sample t test: D t D t / 2sD sD PSYC 6130, PROF. J. ELDER 61 Repeated Measures Designs • Many matched sample designs involve repeated measures of the same individuals. • This can result in carry-over effects, including learning and fatigue. • These effects can be minimized by counter-balancing the ordering of conditions across participants. PSYC 6130, PROF. J. ELDER 62 Assumptions of the Matched t Test • Normality • Independent random sampling (within samples) PSYC 6130, PROF. J. ELDER 63