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# Scope and Methods POL 242

Hope College

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This 7 page Class Notes was uploaded by Joseph O'Hara on Monday October 12, 2015. The Class Notes belongs to POL 242 at Hope College taught by Joel Toppen in Fall. Since its upload, it has received 12 views. For similar materials see /class/222145/pol-242-hope-college in Political Science at Hope College.

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Date Created: 10/12/15

Module 3 Comparison of Means Test Introduction In the previous module you learned to analyze the relationship between two categorical variables Now you will learn to analyze a relationship where the dependent variable is measured at the interval level and the independent variable is categorical This procedure is known as a comparison of means ttest The theory behind this test is that if we divided the data into groups based on the independent variable the mean value of the dependent variable will be the same for each group Therefore the test looks for differences in the means of each group and checks to see if that difference is statistically significant that is whether the difference could be attributed to chance The independent variable can only have two values thus creating two groups for comparison If the categorical variable has more than two categories it is not suitable for a ttest the proper test for this situation is ANOVA which is not covered in this guide Stata will return an error if you try to perform a ttest with a variable that has more than two groups Using your knowledge of transforming variables you should be able to turn any categorical variable into the proper format for a ttest T T est Hypotheses Since we are comparing two groups there are 3 possible hypotheses we need to consider for a ttest The first hypothesis is that the mean of groupl is larger than the mean of group2 A second possible hypothesis is that the mean of groupl is smaller than the mean of group2 The third possible hypothesis is that there is no difference between the mean of groupl and the mean of group2 this is also the null hypothesis Stata expresses the hypotheses in terms of the difference between the means of the groups Therefore the hypotheses are shown in the following notation H0 mean of groupl 7 mean of group2 difference 0 This is the notation for the null hypothesis Since the difference is equal to 0 there is no difference in the means of the groups Ha diff gt 0 This is one of the alternative hypotheses It says the difference is greater than 0 which means that the mean of groupl must be greater than the mean of group2 Ha diff lt 0 This is the other alternative hypothesis It says that the difference is less than 0 so the mean of groupl must be less than the mean of group2 Stata also includes another alternative hypothesis Ha diff 0 This hypothesis says that the groups have different means but doesn t specify the direction of the difference Since the other two alternative hypotheses do specify a direction we will focus on them and ignore Ha diff 0 32 Example 1 As With the chisquare test of signi cance the best Way to learn to perform a t test is to Work through an example We Will start With an example using some variables from WORLDdta This data set contains a few good examples of every type of data on most of the nations in the World For this example regime is the independent variable and Womleg is the dependent variable Regime is a categorical variable With 0 meaning the nation is a democracy and 1 meaning it is a dictatorship Womleg is an interval variable that measures the percentage of the nation s legislators Who are Women The hypothesis for the relationship is In comparing nations those Who are democracies Will have a higher percentage ofWomen legislators than Will dictatorships In statistical notation our hypothesis Would be Ha diffgt 0 because We expect mean ofWomleg for democracies 7 mean of Womleg for dictatorships to be greater than 0 Graphing a Comparison of Means As With the previous modules it is a good idea to get a visual picture ofthe relationship you are analyzing The best Way to do this for a comparison ofmeans is With chart Since We Want the height of the bars to represent the mean of Womleg We use a bar chart rather than a histogram Which measures frequencies To create the bar chart go to GraphicsgtBar chart The Window that opens is very intimidating but We only need to enter one variable name Womleg in the rst Variable eld Figure 31 Then click the By tab and enter the independent variable regime in the proper eld Figure 32 To label the bars click the Labels and under Label type select Bar Click OK to view the chart Figure 33 bar 7 Hal challs Mam 1m W W magmas ltahelsMrsc umpire IEavlmnImgendIDvevaH Slahshc New name lavhanal Variable New name mammal Variable 91g UK Figure 3 1 33 Main Luge a w I elgml r Enumnlli Ha r 7 Law Delaull v l Ema aim Figure 3 z m democracy dlctatorshlp man a 83225 Graphs by l5 country democracy ur mctatursmm Figure 3 3 The chart shows a difference of over 5 percentage points in the percentage of legislators who are Women in democracies compared to dictatorships Now We need to determine if the difference is statistically signi cant The ttest Will provide the answer 34 Performing a TrTest 39 39 39 39 cahlp amp testsgtclassical tests ofhypothesisgtCrroup mean comparison test Enter the dependent 39 39 ariaure name 39 39 39 39 the r 39 39 chqum variances r 39 39 39 u u r r r 39 39 1 or not we will always assume that th aren t This is a less stringent test than assuming the variances are equal and click OK The default for the Con dence level is 95 which is what we want since we are looking for a value of 05 so don t change it us I Mam hymth Valnbln Mm Ewan vamNA Ham warmed legals l7 Unswsl valance l Welch39rapwwmamn 553 Ennridanc level m Figure 34 quot39 H19 i m c r val eaui Additinnall eveml 39 each group Figure 35 35 From this table we can see that the mean percentage of women legislators is 1419 in democracies and only 883 in dictatorships The average difference between these means is 536 How do we know if this is a statistically signi cant difference What should we decide about the hypotheses Again the answer lies in the Pvalue The Pvalue for a ttest is determined much like the Pvalue for the chisquare test is There is a distribution of the critical values for the ttest see end of guide Using the tstatistic produced by Stata using an intricate mathematical formula involving the means counts and variances and the degrees of freedom produced by Stata or generally the count of the lowest group minus 2 for the test you can find the Pvalue Stata produces the exact Pvalue for you For our example we find our hypothesis in the output bottomright of the screen and read that it had a tstatistic of 30491 which yields a Pvalue of 00016 This tells us that we would observe a difference in the data as large as we did only 016 of the time by chance Thus there is strong evidence that the mean of womleg is not the same for democracies and dictatorships Furthermore the test supports the hypothesis that democracies have a higher percentage of women legislators than dictatorships do Example 2 Let s do another example and combine ideas from this module and module 1 together We will use STATESdta for this test Our hypothesis is In comparing states those in the west region will have a higher rate of unemployment than will states in the other regions To measure unemployment STATESdta contains a variable unemploy which measures a state s level of unemployment in percents and is suitable for our independent variable The data set does not contain a variable that separates states into those in the west and those not in the west but it does contain region a variable that assigns 1 to states in the northeast 2 to those in the midwest 3 to those in the south and 4 to those in the west Since a ttest requires the independent variable to have only two groups we need to recode this variable We did this in module 1 but to refresh your memory go to DatagtCreate or change variablesgtOther variable transformation commandsgtRecode categorical variable Enter region in the Variables field The syntax for the first recode is 13 0 Not in West and the second is 4 1 In West This code will assign a 0 to all states in regions not in the west and assign Not in West as their value label All states in the west are assigned a 1 and labeled In West See Figure 36 to make sure you have this correct Make sure click the Options tab and save the results in a new variable named westdum We have just created a dummy variable that tells us whether a state is in the west region 36 Mam 1m 1 mm Vauah es vegmn Ehaase Me Immals m 2de m Emma yam awn Mes Examp es Rammed m3 n Nat nWeSl Q Q UK Ban22 Suhmu Figure 3 5 tables Figure 3 7 individually 39L states in the west region for each variable so the recode is correct the nation A bar chart will help us to visualize this You should be able to do this on 37 your own but an example ofa possible final chart is provided Figure 3 8 From the NmmWesi Regmn nWestREEian onare w REEDDEaiieviaMcemus remquot Figure 38 chart i uu difference statistically sigiificant or could ithave occured by chance7quot n d e form a trtest to answer the question of statistical sigiificance Go to StatisticsgtSummane tables amp testsgtClassiCal tests of mean compa o s Sr hypothesisgtGroup nson test and fill out the Window Figure 3 gt i nesl mun mean cnmv suns esls Main wryM war a a awe l memau uneaaai one r were name an 95 canine eieYei Iguana vaii h ah ame wesldum Figure 39 Fi ime 0 since we aipect mean of not in westemean ofwest to be less than 0 meaning thatthe westem states have a higherunemploymmtra e 38

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