Chapter 8 Notes
Chapter 8 Notes PSYCH 2220 - 0020
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This 13 page Class Notes was uploaded by Emma Dahlin on Tuesday October 20, 2015. The Class Notes belongs to PSYCH 2220 - 0020 at Ohio State University taught by Anna Yocom in Summer 2015. Since its upload, it has received 27 views. For similar materials see Data Analysis in Psychology in Psychlogy at Ohio State University.
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Date Created: 10/20/15
Con dence Intervals Point estimate summary statistic one number as an estimate of the population 0 Ex mean Interval estimate based on our sample statistic range of sample statistics we would expect if we repeatedly sampled from the same population 0 Con dence interval Con dence intervalsljlnterva estimate that includes the mean we would expect for the sample statistic a certain percentage of the time were we to sample from the same population repeatedly 0 Typically set at 95 The range around the mean when we add and subtract a margin of error Con rms ndings of hypothesis testing and adds more detail 0 If you run a 95 con dence interval it should match results of hypothesis test at 05 likewise if you run a 99 Cl it should match the results of hypothesis test at 01 Calculating Con dence Intervals Step 1 Draw a picture of a distribution that will include con dence intervals use sample mean Step 2 Indicate the bounds of the CI on the drawing based on CI Step 3 Determine the 2 statistics that fall at each line marking the middle 95 Step 4 Turn the 2 statistic back into raw means Step 5 Check that the Cls make sense Example IQ Scores IQ scores are designed to have a mean of 100 and a standard deviation of 15 A school psychologist is convinced that the mean IQ score of the high school seniors in her district is different from 100 She administered an IQ test to random sample of 50 seniors in her district and found their mean IQ was 104 o 95 Cl want to get a range of values and see if it matches up with 100 Look up 25 in table to get 2 scores of 196 and 196 475 475 254 25 1 232 1 0 How is 99 CI different 0 It would include more values in that interval 0 Probability would be higher that true mean would be included Step 4 Calculate raw means 0 Mower 39ZOM Msampe Mupper ZOM Msampe Remember to use sample means amp standard error Mlower 1962121049984 Mupper 196212 10410816 These are values that would mark off upper and lower bounda es Step 5 is sample mean in the middle right in bt 2 values 0 104 is exactly in between 9984 amp 10816 95 Cl is 998410816 p 9984 Sp510816 95 o The probability is 95 that an interval such as 9984 to 10816 contains the true average IQ score 0 Her district could possibly have IQ score of 100 possible they could have same exact mean 0 This would tell us to FAIL TO REJECT NULL bc they could be the same aka not different p is fixed not a variable it s either in the CI or it isn t 0 Only interval varies from experiment to experiment We never get to know whether it s actually in there 0 We just tell the probability given our sample data OOOOO Con dence Intervals 0 Why do we need this It tells us the interval where it is most likely that the true population mean will fall and then we can compare the known one Also tells us whether we should reject or fail to reject our hypothesis O MupperIOWGF Msampe Effect Size Just how big is the difference Signi cantBig 0 Does not mean there is a big difference we cannot tell Signi cantDifferent 0 Not necessarily the case lncreasing sample size will make us more likely to nd a statistically signi cant effect 0 But statistical signi cance does not mean practical signi cance Effect size Size of a difference that is unaffected by sample size 0 Standardization across studies Imagine both represent signi cant effects Which effect is bigger Bottom one Note the spread in the two gures Which effect size is bigger 0 bottom one variability has decreased less overlap Increasing Effect Size Decrease the amount of overlap between two distributions 1 Their means are farther apart 2 The variation within each population is smaller Calculating Effect Zie 0Cohen s destimates effect size oAssesses difference between means using standard deviation instead of standard error MM G d TABLE Cohen s Conventions for Effect Elias d Jami Guitar has lzlllbilsna gudellnes i lquot callquotielrlialrlsiu based Elli the overlap baf esan tab flisfrlbutbrls to help researchers determine whether an effect is small medium Elf large These numbers are nut cutoffs merely rrlLIgll guidelines iii Ellil researchers in their interpretation of results Effect Site Content Marian Small 12 35 lJlatiliulfl 13 WEI El Large 03 31392 Calculating Effect Size Example IO Scores d 10410015O27 We are saying that there is a lot of overlap bt these 2 groups 0 Degree to which the participants mean exceeded the value expected by chance 0 The average score of the participants was 027 SD higher than the population mean 0 Based on Cohen s guidelines this would be considered a small effect 0 Effect size tells you nothing about statistical signi cance Even if you have a large effect it might not be statistically signi cant Statistical Power 0 Measure of our ability to reject null hypothesis given that null is false The probability we will 0 Reject null when we should 0 Find an effect difference when it really exists 0 Avoid Type II error 3 so power 1 3 o In hypothesistesting we compare two states of the world Q True and g False can be represented as two distinct patterns of normal distributions 0 Alpha or cuts through the g 77quotue distributions but also maps onto a point in the H0 False distributions for a given effect size partitioning Band 1B So as alpha decreases power decreases 0 Just not in 11 proportion tierisi39tfjjf A pine Power a n d Type lli Er fore Zens uniier null WI I l i i l a Z 2 d If iesi statistic In a hypothesis test we need to know the distribution of our test statistic when assuming the null hypothesis is true The rejection region is an interval with area or under the null distribution curve Hence if the null hypothesis is true we will reject Type I error with probability a A power calculation determines how likely we are to reject if the null hypothesis is not true Suppose the population mean is u3 In this case the distribution of the test statistic follows the red curve The area of the rejection region under the true red distribution is the probability that we will reject This is called power The rest of the area under the true red distribution is the probability that we fail to reject Type II Error Hence Power 1 PrType ll Error If the difference between the null and true distributions is small then the test has low power If the test statistic has less variability then the distributions do not overlap as much Hence there is more power If we lower or then we require more evidence to reject Therefore the rejection region is smaller and we have less power Alpha In uences Power Power a e 5 00 Pewe r b l 1 0 00 TwoTailed or OneTailed Test In uences Power Fewer l a lifi 2500 2500 iner in 39 i 5 Sample Size and Standard Deviation In uence Power P owe r a 5 P Owe r 5 b std error we have direct control over this arger sample size on bottom smaller std deviation Difference between the Means In uences Power Power a LI l 5 09 Power 5 00 Factors Affecting Power 1 Alpha evehigher alpha increases power 0 Potential problem Increase change for Type I Error 2 One or twotailed test 1taiIed test increases power 0 Potential problem Only helpful if CERTAIN of direction of effect 3 Sample Size and Variability Larger sample size and smaller standard deviation less variability reduces noise and increases power 4 Actual difference effect size 0 Increase difference bt means stronger manipulationmore pronounced effect What we do with Power 0 Just like we choose our preferred risk of Type I error we d like to choose our power to detect rea effects 0 But we can only directly in uence the factors mentioned 0 For power calculations we typically 0 Calculate sample size needed for a target power level pO80 or above 0 Calculate the power available at a target sample size 0 Calculate what size of effects we can detect with speci ed power amp sample size Power amp Effect Size What s the Point 0 We want to build research that has a good chance of being successful correctly rejecting Ho That is we want studies with high power o If we have a good guess about effect size then we can calculate how many participants N our research will need in order to be 90 sure of correctly rejecting H0 0 I want to have 90 power to detect real effects that are at least moderate in size dO5 How many participants will I need to recruit for my study Alternately we can gure out how likely research with a speci c size N is to demonstrate a true effect 0 I only have access to 56 stats students for an educational intervention Assuming 28 people in each of two groups how much power will I have to demonstrate a moderate size effect 0 We can even understand how high power lets us draw some inferences we normally wouldn t o If we have 99 power to detect even a very small effect size and then I don t detect itwel the effect size is probably zero 0 That is can draw a reasonable inference that H0 is true 0 Requires appropriate research design and the very large N needed for such high power NOTES FROM TEXTBOOK 1 Compute con dence intervals which provide range of plausible mean differences 2 Calculate effect sizes which indicate the size of differences 3 Estimate statistical power of study to be sure that we have suf cient sample size to detect a real difference 0 Point estimateljsummary statistic from a sample that is just one number used as an estimate of the population parameter 0 Rarely exactly accurate Interval estimateljbased on a sample statistic and provides a range of plausible values for the population parameter 0 Usually found by addingsubtracting margin of error from point estimate 0 Con dence intervalljan interval estimate based on the sample statistic it includes the population mean a certain percentage of the time if we sample from the same population repeatedly o 95 con dence level is most commonly used 0 Con dence interval is range bt the two values that surround the sample mean 0 When 2 intervals do not overlap we conclude that the population means are likely different CALCULATING CONFIDENCE INTERVALS 1 Draw a picture of a distribution that will include the con dence interval ldraw normal curve w sample mean at center not population mean 2 Indicate the bounds of the con dence interval on the drawing lfor a 95 con dence interval there should be 25 in each tail 3 Determine the 2 statistics that fall at each line marking the middle 95 luse 2 table for 95 con dence interval 2 statistics of 196 and 196 4 Turn the 2 statistics back into raw meansuse sample mean and standard err0rbc it is a distribution of means nd raw mean for both the upper and lower 2 scores 5 Check that the con dence interval makes sense lsample mean should fall exactly in middle of two ends of the interval o If you sample the same population over and over again the 95 con dence interval would include the population mean 95 of the time Because a statistically signi cant effect might not be an important one we should calculate effect size to tell us whether statistically signi cant effect is small medium or large Statistically signi cantquot does not mean that the ndings from a study represent a meaningful difference 0 Means that ndings are unlikely to occur if the null hypothesis is true 0 Effect size tells us whether a statistically signi cant difference might also be an important difference 0 Increasing sample size always increases the test statistic and decreases sta nda rd error 0 Large samples allow us to more readily reject the null hypothesis 0 Effect sizeljlindicates the size of a difference and is unaffected by sample size 0 Tes us how much two populations do not overlap 0 Small overapbiggereffect size 0 Standardized score that can allow you to compare scores across different studies 0 Effect size can be decreased when means are further apart and when variability Within each distribution of scores is smaller 0 When we conduct a 2 test the effectsize statistic is typically Cohen s d Cohen s dljlmeasure of effect size that assesses the difference bt 2 means in terms of standard deviation not standard error 0 Small effect O285 overlap 0 Medium effect O567 overlap 0 Large effect O853 overlap It is magnitude of effect size that matters effect size of O5 is same size as 05 Metaanalysisa study that involves the calculation of a mean effect size from the individual effect sizes of many studies 0 Provides added statistical power by considering many studies simultaneously STEPS 1 Select topic of interest and decide exactly how to proceed before beginning to track down studies 2 Locate every study that has been conducted and meets criteria 3 Calculate an effect size often Cohen s d for every study 4Cacuate statistics ideally summary statistics a hypothesis test a con dence interval and a visual display of the effect snzes Review of concepts 0 As sample size increases the test statistic becomes more extreme and it becomes easier to reject the null hypothesis 0 A statistically signi cant result is not necessarily one with practical importance Effect sizes are calculated with respect to scores rather than means so they are not contingent on sample size The size of an effect is based on the difference bt two group means and the amount of variability within each group Effect size for a ztest is measured with Cohen s d which is calculated much like a zstatistic but using standard deviation instead of standard error A metaanalysis is a study of studies that provides a more objective measure of an effect size than an individual study does Statistical powerljla measure of the likelihood that we will reject the null hypothesis given that the null hypothesis is false 0 Probability that we will reject the null when we should reject it 0 Probability that we will not make a ijpe error Calculation of statistical power ranges from probability of 000 to probability of 100 0 to 100 o Ideally a researcher only conducts a study when there is 80 statistical power that is at least 80 of the time the researcher will correctly reject the null hypothesis STEPS 1 Determine information needed for calculationhypothesized mean for the sample sample size population mean population standard deviation and standard error based on sample size 2 Determine critical value in terms of 2 distribution and the raw mean so that statistical power can be calculated 3 Calculate the statistical power the percentage of the distribution of means for population 1 the distribution centered around the hypothesized sample mean that falls above the critical value On a practical level statistical power calculations tell researchers how many participants are needed to conduct a study whose ndings we can trust FIVE FACTORS THAT AFFECT STATISTICAL POWER 1 Increase alpha 2 Turn twotailed hypothesis into a onetailed hypothesis 3 Increase N 4 Exaggerate mean difference bt levels of the independent variable 5 Decrease standard deviation
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