Action Research T-tests INFO 515 Glenn Booker INFO 515 Lecture #5 1 Statistical Significance A little review before we discuss T tests The Alternative, or Research, Hypothesis states the researcher’s belief that some difference or effect exists The Null Hypothesis states that no effect or no difference exists in the data INFO 515 Lecture #5 2 Significance versus importance Keep in mind that the results of a test may be statistically significant, but that doesn’t mean the results are important INFO 515 Just because most serial killers ate cereal as a child doesn’t mean eating cereal makes one a serial killer (just a cereal killer) “Correlation doesn’t imply causation” is a similar saying Lecture #5 3 Confidence Intervals For a large population, N, the confidence interval for the mean is given by the mean +/- the critical z score (zc, e.g. 1.96 for 95% confidence level) times the standard error CI = X ± zc(SEx) where SEx = s/ (n) INFO 515 Lecture #5 4 Confidence Intervals – Small N Correct for a small population Np by using CI = X ± {zc(SEx)* [ (Np-n)/(Np-1) ] } where ‘n’ is the sample size This is an alternative to using the critical t value Schaum has a summary of zc for various confidence levels from 50% to 99.73% on page 203, Table 9-1 (3rd Ed.) zc depends only on confidence level, not N INFO 515 Lecture #5 5 Probable Error The probable error of some parameter S is the same as the 50% confidence interval CI = X ± 0.6745(SEx) where the variables are the mean and standard error terms INFO 515 Lecture #5 6 Compare Means Using T Tests There are three major types of T tests; all are used to compare means of one data set to a fixed value, or to another data set “One-sample T Test” (SPSS phrasing) By hand, we used the Student T test from last week for small sample sizes (n<30) Uses z scores for large sample sizes “Independent-samples T Test” Dependent- or “Paired-samples T Test” In SPSS, see Analyze / Compare Means INFO 515 Lecture #5 7 Check Data First Before using any T test, it is helpful to generate a boxplot of the data set Since we’re looking at means, look for extreme values or outliers which may distort the mean May want to hide extreme values INFO 515 In SPSS, use Data > Select Cases… Lecture #5 8 Example T Test Null Hypotheses One-sample Compare one sample’s mean to a normal, typical, or population value Does Sample average = the normal value? Independent INFO 515 Compare means of two samples which don’t directly influence each other (samples are two different groups of people or things) Does average income for Midwesterners = average income for Southerners? Lecture #5 9 Example T Test Null Hypotheses Dependent (or Paired) INFO 515 Compare means of two samples which you expect to be connected (often, data is from the same sample at two different times) Does Average productivity before training = average productivity after? Lecture #5 10 One-Sample T-Test This variation compares one sample mean against a mean derived from an independent source, for example a published source Are this class’ average IQ scores substantially different from the average of 100? Does the average income in my development differ from the average for my city? The “mean derived from an independent source” is called the Test Value in SPSS INFO 515 Lecture #5 11 One-Sample T Test The One Sample T Test compares a mean to a fixed value (which is often representative of the population, or an ideal value) For this example, use the data file, world95.sav INFO 515 Lecture #5 12 One-Sample T Test Given that the birth to death rate (b_to_d) should be under 1.25 for a country’s population to remain stable Identify whether countries with different predominant religions have average birth to death rates different from this value But “religion” is a string variable with many specific values - how handle this? INFO 515 Could Select Cases for each religion separately, but that’s tedious! Lecture #5 13 One-Sample T Test Instead, isolate three religions using Data / Select Cases Reset the values Choose If with the condition Note that “|” means “or” (an artifact from some programming languages), and double quotes are used in the expression, not single INFO 515 religion = “Protstnt” | religion = “Catholic” | religion = “Muslim” Earlier versions of SPSS used single quotes! Lecture #5 14 One-Sample T Test Use Data / Split File... Choose to “Compare Groups” Select “Predominant religion” as the “Groups Based on:” criterion In the Data View, note that the “Filter On” and “Split File On” messages are displayed in the lower right status bar Now generate the actual analysis... INFO 515 Lecture #5 15 One-Sample T Test Use Analyze / Compare Means / OneSample T Test… / Select Birth to Death Ratio as the Test Variable(s) Set the Test Value to 1.25 Use the Options button to set the Confidence Interval to 99% instead of the default 95% INFO 515 Lecture #5 16 One-Sample T Test e E e N e e P C B 1 5 9 6 M B 7 2 9 5 P B 6 6 8 7 m l u o n a l e r e a e o 2 p d P t r w p f e C B 4 0 0 5 6 5 M B 0 6 0 2 0 5 P B 3 5 1 6 3 4 INFO 515 Lecture #5 17 One-Sample T Test Interpretation of the output is exactly the same for all T Tests Reject the null hypothesis (here, b_to_d distribution includes the value 1.25) if INFO 515 The magnitude of ‘t’ is greater than the critical two-tailed t value (which you’d look up) The significance (Sig.) is under 0.010 (for 99% confidence) The confidence interval does not include zero Lecture #5 18 One-Sample T Test Note that all three cases would have birth to death ratios different from 1.25 at the 95% level of confidence (reject the null hypothesis for all three religions) But at a 99% confidence level, the Protestant countries accept the null hypothesis INFO 515 Significance is over 0.010 Confidence interval includes zero ‘t’ value is under 2.947 (which is tc for two-tail at 99% confidence (Sig = .01)) Lecture #5 19 Independent Sample T-Test This variety of t-test compares two independent means Two groups are measured using the same instrument (device, technique), for example: INFO 515 Two groups of people who were asked the same survey Two samples of parts which were measured with the same tool Lecture #5 20 Independent Sample T-Test Other examples INFO 515 Compare means of two groups of people (average income for Democrats vs Republicans) Compare means for different types of patrons (average library usage for graduate students vs undergrad) Lecture #5 21 Independent Samples T Test With SPSS, use Analyze / Compare Means / Independent-Samples T Test… / Select the Test Variable(s), which are the variables whose means you want to examine (test score, income, etc.) The Grouping Variable is the distinguishing characteristic you want (gender, political party, climate,…) INFO 515 Lecture #5 22 Independent Samples T Test Key limitation for the Independent Samples T Test INFO 515 You have to specify exactly two values for the Grouping Variable, if more than two are possible (otherwise this is the ANOVA lecture – analysis of variance) Lecture #5 23 Independent Samples T Test Suppose we want to know if the average daily calorie intake is different for people who live in tropical climates, than those in temperate climates (hypothesis) For the “world95.sav” data set, INFO 515 use Analyze / Compare Means / IndependentSamples T Test… / Use ‘Daily calorie intake’ as Test Variable Grouping Variable ‘predominant climate’ = tropical and temperate (#5 and 8), we get: Lecture #5 24 Indep. Samples T Test Inputs INFO 515 Lecture #5 25 Independent Samples T Test Group Statistics Daily calorie intake Predominant climate tropical temperate N Mean Std. Deviation Std. Error Mean 28 2374.93 308.809 58.359 23 3216.65 529.417 110.391 S s T u f a V o n v a l e r M . e E a S e 2 o e F d p i t r g w r f p e D E 8 0 7 2 9 0 2 6 8 7 a s E 9 1 1 0 2 7 2 2 n INFO 515 Lecture #5 26 Independent Samples T Test The Group Statistics section gives general descriptive statistics for the data Note that the sample size, mean, standard deviation, and standard error are all given The Independent Samples Test section is where this specific test’s results are presented INFO 515 What’s this stuff about “Equal variances”? Lecture #5 27 Levene’s Test Levene’s Test tells whether to use “Equal variances assumed” or not The null hypothesis for Levene’s Test is that “the variances for the two groups of data are equal” Like most of this statistical universe: if the Sig. of the test is below the critical value (0.050), reject the null hypothesis Notice that ‘df’ changes, depending on the outcome of Levene’s Test INFO 515 Lecture #5 28 Levene’s Test So to interpret Levene’s Test: If its Significance level is below 0.050, use the “Equal variances not assumed” row of output; assume separate variances, which reduces df If the Significance level is >= 0.050 use “Equal variances assumed” which is also called “pooled variances” Often the results are the same either way INFO 515 Lecture #5 29 Independent Samples T Test So what’s the answer to this problem? You could look at the ‘t’ value, and compare it to the critical t score for your desired level of confidence (here tc = 2.034) INFO 515 |t| > tc means reject the null hypothesis Or, if the T-Test’s significance level (Sig. (2-tailed)) is below your desired value (e.g. 0.050), then reject the null hypothesis (i.e. there is a significant difference) Lecture #5 30 Independent Samples T Test Could also use the confidence interval for the difference in the means - if it includes zero, then accept the null hypothesis For this example, all three ways to evaluate the results agree INFO 515 We conclude that people in temperate climates do not consume the same amount as those in tropical climates Lecture #5 31 Dependent Sample T-Tests Related sample, paired sample, correlated sample--these are all names for dependent means T-Tests (SPSS calls them Paired-Samples) INFO 515 Each subject in one sample has a corresponding subject in the second sample… often the same person, or organization, or whatever thing the subject is Lecture #5 32 Dependent Sample T-Tests Examples INFO 515 Mean test scores for one group of students, before and after receiving training Mean pulse, blood pressure, or some other body characteristic before and after receiving some medication Mean IQ scores for the same group of people, for two consecutive years Lecture #5 33 Dependent Sample T-Tests This compares the averages of two columns of data, which generally relate to before and after some treatment Can also be used to compare two related variables INFO 515 Here, use “world95.sav” to compare the life expectancy for males and females in OECD countries (the USA and Western Europe) Male vs. female sounds like a weak example, but wait a few slides and we’ll show it isn’t Lecture #5 34 Dependent Sample T-Tests Use Data / Select Cases to isolate If region=1 In SPSS, use Analyze / Compare Means / Paired-Samples T Test… / INFO 515 Click on “lifeexpf” and “lifeexpm”, then move them both to the Paired Variables section at once (you have to select two variables) The Options button allows you to change the default 95% confidence level (but don’t) Lecture #5 35 Dependent Sample T-Tests l E e N e e a P A 0 1 8 6 1 l A 1 1 5 5 l e s e N i l P A 1 e 4 0 m INFO 515 Lecture #5 36 Dependent Sample T-Tests The Paired Samples Statistics shows the general descriptive statistics for the data The Paired Samples Correlations section indicates whether there is a strong relationship between the data A small significance (<0.050) indicates that there is a correlation, and hence a paired test is better than a test for independent variables INFO 515 This justifies using the male/female example as dependent samples! Lecture #5 37 Dependent Sample T-Tests Paired Samples Test Paired D ifferences Mean Pair 1 Average female life expectancy - Average male life expectancy INFO 515 6.381 Std. Deviation .865 Std. Error Mean .189 Lecture #5 95% C onfidence Interval of the Difference Lower 5.987 Upper 6.775 t 33.819 df Sig. (2-tailed) 20 .000 38 Dependent Sample T-Tests The Paired Samples Test section gives the actual test results The Paired Differences describes the distribution of the differences between pairs of data We have three ways to determine the outcome; reject the null hypothesis if: t > critical t score for two-tailed test (here tc=2.086) Sig. (2-tailed) < 0.050 (desired significance level) Confidence interval for the difference does not include zero Psst! INFO 515 Is this pattern looking familiar yet? Lecture #5 39 One-tailed and two-tailed tests Choosing an appropriate test - one-tailed or two-tailed - will depend on the research hypothesis A “Difference” in the hypothesis implies no specific direction: use two-tailed test Only “Greater than” hypotheses or “less than” hypotheses imply a one-tailed test INFO 515 This is a weaker assumption, and results in a smaller, one-directional, critical t value Lecture #5 40 One or Two Tails? We have used two tail tests so far, to see if the likely range of some parameter (e.g. mean), is different from another (could be either larger or smaller) A one tailed test is used to check if something is larger than another OR something is less than another (pick just one of those options) INFO 515 Lecture #5 41 One or Two Tails? The effect of using the one tail test is that we want the entire left or right side of the distribution to contain all of the uncertainty INFO 515 One-tail lowers zc because the test only checks one side of the distribution, hence all of the uncertainty is on that side of the distribution For example, zc for one-tailed confidence of 95% is the same as zc for two-tailed at 90% Lecture #5 42 One or Two Tails? C onfidence lev el 95% 99% O ne tailed z c -1.645 or + 1.645 -2.33 or + 2.33 T w o tailed z c -1.96 and + 1.96 -2.57 and + 2.57 (Schaum p. 218) INFO 515 Lecture #5 43 One or Two Tails? The bottom line? As a rule of thumb, use two-tailed tests unless the circumstances being measured make it physically impossible for a twotailed outcome to occur INFO 515 Lecture #5 44

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