Statistiek II. John Nerbonne. October 1, Dept of Information Science


 Ezra Wade
 2 years ago
 Views:
Transcription
1 Dept of Information Science October 1, 2010
2 Course outline 1 Oneway ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic regression.
3 Lecture outline Today: Oneway ANOVA 1 General motivation 2 F test and F distribution 3 ANOVA example 4 The logic of ANOVA Short break 5 ANOVA calculations 6 Posthoc tests
4 What s ANalysis Of VAriance (ANOVA)? Most popular statistical test for numerical data Generalized ttest Compares means of more than two groups Fairly robust Based on F distribution compares variances (between groups and within groups) Two basic versions: a Oneway (or single) ANOVA: compare groups along one dimension, e.g., grade point average by school class b Nway (or factorial) ANOVA: compare groups along 2 dimensions, e.g., grade point average by school class and gender
5 Typical applications Oneway ANOVA: Compare time needed for lexical recognition in 1. healthy adults 2. patients with Wernicke s aphasia 3. patients with Broca s aphasia Factorial ANOVA: Compare lexical recognition time in male and female in the same three groups
6 Comparing multiple means For two groups: use ttest Note: testing for pvalue of 0.05 shows significance 1 time in 20 if there is no difference in population mean (effect of chance) But suppose there are 7 groups, i.e., we test ( 7 2) = 21 pairs of groups Caution: several tests (on the same data) run the risk of finding significance through sheer chance
7 Multiple comparison problem Example: Suppose you run k = 3 tests, always seeking a result significant at α = 0.05 probability of getting at least one false positive is given by: α FW = 1 P(zero false positive results) = 1 (1 α) k = 1 (1 0.05) 3 = 1 (0.95) 3 = Hence, with only 3 pairwise tests, the chance of committing type I error almost 15% (and 66% for 21 tests!) α FW called Bonferroni familywise αlevel
8 Bonferroni correction for multiple comparisons To guarantee a familywise αlevel of 0.05, divide α by number of tests. Example: 0.05/3 (= α/# tests) = (note: ) set α = (= Bonferronicorrected αlevel) If pvalue is less than the Bonferronicorrected target α: reject the null hypothesis. If pvalue greater than the Bonferronicorrected target α: do not reject the null hypothesis.
9 Analysis of variance ANOVA automatically corrects for looking at several relationships (like Bonferroni correction) Based on F distribution: Moore & McCabe, 7.3, pp Measures the difference between two variances (variance σ 2 ) F = s2 1 s 2 2 always positive since variances are positive two degrees of freedom interesting, one for s 1, one for s 2
10 F test vs. F distribution F value: F = s2 1 s 2 2 F values used in F test (Fisher s test) H 0 : samples are from same distribution (s 1 = s 2 ) H a : samples are from different distributions (s 1 s 2 )  value near 1 indicates same variance  value near 0 or + indicates difference in variance F test very sensitive to deviations from normal ANOVA uses F distribution, but is different: ANOVA F test!
11 F distribution Critical area for F distribution at p = 0.05 (df: 12,10) Probability density rejection region acceptance region rejection region Note the symmetry: P( s2 1 s 2 2 F value < x) = P( s2 2 s 2 1 (because y < x 1 y > 1 x for x, y R + ) > 1 x )
12 F test Example: height group sample mean standard size deviation boys cm 6cm girls 9 168cm 4cm Is the difference in standard deviation significant? Examine F = s2 boys s 2 girls Degrees of freedom: df boys = 16 1 df girls = 9 1
13 F test critical area (for twotailed test with α = 0.05) P(F (15, 8) > x) = α 2 = P(F (15, 8) < x) = P(F (15, 8) < 4.1) = Moore & McCabe, Table E, p. 706 P(F (15, 8) < x) = P(F (8, 15) > x ) = where x = 1 x P(F (8, 15) > 3.2) = (tables) P(F (15, 8) < ) = P(F (15, 8) < 0.31) = (no values directly for P(F (df 1, df 2 ) > x)) Reject H 0 if F < 0.31 or F > 4.1 Here, F = = 2.25 (hence no evidence of difference in distributions)
14 ANOVA Analysis of Variance (ANOVA) most popular statistical test for numerical data several types  single, oneway  factorial, two, three,..., nway  single/factorial repeated measures examines variation  betweengroups gender, age, etc.  withingroups overall automatically corrects for looking at several relationships (like Bonferroni correction) uses F distribution, where F (n, m) fixes n typically at the number of groups (minus 1), m at the number of subjects, i.e., data points (minus number of groups)
15 Detailed example: oneway ANOVA Question: Are exam grades of four groups of foreign students Nederlands voor anderstaligen the same? More exactly, are the four averages the same? H 0 : µ 1 = µ 2 = µ 3 = µ 4 H a : µ 1 µ 2 or µ 1 µ 3... or µ 3 µ 4 Alternative hypothesis: at least one group has a different mean For the question of whether any particular pair is different, the ttest is appropriate. For testing whether all language groups are the same, pairwise ttests exaggerate differences (increase the chance of type I error) We therefore want to apply oneway ANOVA
16 Data: Dutch proficiency of foreigners Four groups of ten students each: Group Europe America Africa Asia Mean Samp. SD Samp. Variance
17 ANOVA conditions ANOVA assumptions: Normal distribution per subgroup Same variance in subgroups: least SD > onehalf of largest SD independent observations: watch out for testretest situations! Check differences in SD s! (some SPSS computing)
18 ANOVA conditions Assumption: normal distribution per group, check with normal quantile plot, e.g., for Europeans below (repeat for every group) Normal QQ plot of toets.nl voor anderstalige
19 Visualizing ANOVA data Is there a significant difference in the means (of the groups being contrasted)? Take care that boxplots sketch medians not means.
20 Sketch of ANOVA Group Eur. Amer. Africa Asia x 1j x 2j x 3j x 4j x 1 x 2 x 3 x 4 Notation: Group index: i {1, 2, 3, 4} Sample index: j N i = size of group i Data point x ij : ith group, jth observation Number of groups: I = 4 Total mean: x Group mean: x i For any data point x ij : (x ij x) = (x i x) + (x ij x i ) total residue = group diff. + error ANOVA question: does group membership influence the response variable?
21 Two variances Reminder of high school algebra: (a + b) 2 = a 2 + b 2 + 2ab ab a 2 a b 2 ab b
22 Two variances Data point x ij : (x ij x) = (x i x) + (x ij x i ) Want sum of squared deviates for each group: (x ij x) 2 = (x i x) 2 + (x ij x i ) 2 + 2(x i x)(x ij x i ) Sum over elements in ith group: N i (x ij x) 2 = j=1 N i N i (x i x) 2 + (x ij x i ) 2 + 2(x i x)(x ij x i ) j=1 j=1 j=1 N i
23 Two variances Note that this term must be zero: Because: N i 2(x i x)(x ij x i ) j=1 (a) (b) N i 2(x i x)(x ij x i ) = 2(x i x) (x ij x i ) j=1 N i (x ij x i ) = 0 x i = j=1 N i j=1 } {{ } Ni j=1 x ij 0 N i
24 Two variances So we have: N i j=1 (x ij x) 2 = N i N i (x i x) 2 + (x ij x i ) 2 j=1 j=1 N i (+ 2(x i x)(x ij x i ) = 0) j=1 Therefore: N i j=1 (x ij x) 2 = N i N i (x i x) 2 + (x ij x i ) 2 j=1 j=1 And finally we can sum over all groups: I N i I N i I N i (x ij x) 2 = (x i x) 2 + (x ij x i ) 2 i=1 j=1 i=1 j=1 i=1 j=1
25 ANOVA terminology (x ij x) = (x i x) + (x ij x i ) total residue = group diff. + error I N i I I N i (x ij x) 2 = N i (x i x) 2 + (x ij x i ) 2 i=1 j=1 i=1 i=1 j=1 SST SSG SSE Total Sum of Squares = Group Sum of Squares + Error Sum of Squares (n 1) = (I 1) + (n I ) DFT DFG DFE Total Degrees of Freedom = Group Degrees of Freedom + Error Degrees of Freedom
26 Variances are mean squared differences to the mean Note that SST/DFT: P I P Ni i=1 j=1 (x ij x) 2 n 1 is a variance, and likewise SSG/DFG: labelled MSG ( Mean square between groups ), and SSE/DFE: labelled MSE ( Mean square error or sometimes Mean square within groups ) In ANOVA, we compare MSG (variance between groups) and MSE (variance within groups), i.e. we measure F = MSG MSE If this F value is large, differences between groups overshadow differences within groups.
27 Two variances 1) Estimate the pooled variance of the population (MSE): MSE = DFE SSE PI P Ni = i=1 j=1 (x ij x i ) 2 n I equiv = P I i=1 DF i s 2 i P I i=1 DF i In our example (Nederlands for anderstaligen): P I i=1 DF i s 2 i P I i=1 DF i = (N1 1)s2 1 + (N 3 1)s (N 3 1)s (N 4 1)s 2 4 (N 1 1) + (N 3 1) + (N 3 1) + (N 4 1) = = = Estimates the variance in groups (using DF), aka withingroups estimate of variance
28 Two variances 2) Estimate the betweengroups variance of the population (MSG): MSG = DFG SSG P = I i=1 N i (x i x) 2 I 1 In our example (Nederlands for anderstaligen): We had 4 group means: 25.0, 21.9, 23.1, 21.3, grand mean: 22.8 MSG = 10 (( )2 +( ) 2 +( ) 2 +( ) 2 ) 4 1 = 26.6 The betweengroups variance (MSG) is an aggregate estimate of the degree to which the four sample means differ from one another
29 Interpreting estimates with F scores If H 0 is true, then we have two variances: Betweengroups estimate: s 2 bg = Withingroups estimate: s 2 wg = and and their ratio s 2 bg s 2 wg follows an F distribution with: (# groups 1) = 3 degrees of freedom for s bg 2 and (# observations # groups) = 36 degrees of freedom for swg 2 In our example: F (3, 36) = = 0.55 P(F (3, 40) > 2.84) = 0.05 (see tables), so there is no evidence of nonuniform behavior
30 SPSS summary No evidence of nonuniform behavior
31 Other questions ANOVA H 0 : µ 1 = µ 2 =... = µ n But sometimes particular contrasts are important e.g., are Europeans better (in learning Dutch)? Distinguish (in reporting results): prior contrasts questions asked before data is collected and analyzed post hoc (posterior) questions questions asked after data collection and analysis datasnooping is exploratory, cannot contribute to hypothesis testing
32 Prior contrasts Questions asked before data collection and analysis e.g., are Europeans better (in learning Dutch)? Another way of putting this: H 0 : µ Eur = 1 3 (µ Am + µ Afr + µ Asia ) H a : µ Eur 1 3 (µ Am + µ Afr + µ Asia ) Reformulation (SPSS requires this): H 0 : 0 = µ Eur µ Am µ Afr µ Asia
33 Prior contrasts in SPSS Mean of every group gets a coefficient Sum of coefficients is 0 A ttest is carried out and twotailed pvalue is reported (as usual): No significant difference here (of course) Note: prior contrasts are legitimate as hypothesis tests as long as they are formulated before data collection and analysis
34 Posthoc questions Assume H 0 is rejected: which means are distinct? Datasnooping problem: in a large set, some distinctions are likely to be statistically significant But we can still look (we just cannot claim to have tested the hypothesis) We are asking whether m i m j is significantly larger, we apply a variant of the ttest The relevant sd is MSE n (differences among scores), but there is a correction since we re looking at a proportion of the scores in any one comparison
35 SD in posthoc ANOVA questions Standard deviation (among differences in groups i and j): sd = t = MSE N i +N j N = = 4.9 x i x j sd q 1 N i + 1 N j The critical tvalue is calculated as p c where p is the desired significance level and c is the number of comparisons. For pairwise comparisons: c = ( ) I 2
36 Posthoc questions in SPSS SPSS posthoc Bonferroni searches among all groupings for statistically significant ones But in this case there are none (of course)
37 How to win at ANOVA Note the ways in which the F ratio increases (i.e., becomes more significant): F = MSG MSE 1. MSG increases: differences in means between groups grow larger 2. MSE decreases: overall variation within groups grows smaller
38 Two models for grouped data x ij = µ + ɛ ij x ij = µ + α i + ɛ ij First model: no group effect each data point represents error (ɛ) around a mean (µ) Second model: real group effect each data point represents error (ɛ) around an overall mean (µ), combined with a group adjustment (α i ) ANOVA asks: is there sufficient evidence for α i?
39 Next week Next week: factorial ANOVA
ANOVA  Analysis of Variance
ANOVA  Analysis of Variance ANOVA  Analysis of Variance Extends independentsamples t test Compares the means of groups of independent observations Don t be fooled by the name. ANOVA does not compare
More informationSection 13, Part 1 ANOVA. Analysis Of Variance
Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability
More informationANOVA MULTIPLE CHOICE QUESTIONS. In the following multiplechoice questions, select the best answer.
ANOVA MULTIPLE CHOICE QUESTIONS In the following multiplechoice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard
More informationStatistiek I. ttests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35
Statistiek I ttests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistieki/ John Nerbonne 1/35 ttests To test an average or pair of averages when σ is known, we
More information1.5 Oneway Analysis of Variance
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
More informationANOVA Analysis of Variance
ANOVA Analysis of Variance What is ANOVA and why do we use it? Can test hypotheses about mean differences between more than 2 samples. Can also make inferences about the effects of several different IVs,
More information1 Basic ANOVA concepts
Math 143 ANOVA 1 Analysis of Variance (ANOVA) Recall, when we wanted to compare two population means, we used the 2sample t procedures. Now let s expand this to compare k 3 population means. As with the
More informationChapter 11: Linear Regression  Inference in Regression Analysis  Part 2
Chapter 11: Linear Regression  Inference in Regression Analysis  Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.
More informationTHE KRUSKAL WALLLIS TEST
THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKALWALLIS TEST: The nonparametric alternative to ANOVA: testing for difference between several independent groups 2 NON
More informationINTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the oneway ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
More informationSimple Linear Regression
Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression Statistical model for linear regression Estimating
More informationChapter 5 Analysis of variance SPSS Analysis of variance
Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means Oneway ANOVA To test the null hypothesis that several population means are equal,
More informationAn example ANOVA situation. 1Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)
An example ANOVA situation Example (Treating Blisters) 1Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment
More informationAnalysis of Data. Organizing Data Files in SPSS. Descriptive Statistics
Analysis of Data Claudia J. Stanny PSY 67 Research Design Organizing Data Files in SPSS All data for one subject entered on the same line Identification data Betweensubjects manipulations: variable to
More informationANOVA ANOVA. TwoWay ANOVA. OneWay ANOVA. When to use ANOVA ANOVA. Analysis of Variance. Chapter 16. A procedure for comparing more than two groups
ANOVA ANOVA Analysis of Variance Chapter 6 A procedure for comparing more than two groups independent variable: smoking status nonsmoking one pack a day > two packs a day dependent variable: number of
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationUNDERSTANDING THE TWOWAY ANOVA
UNDERSTANDING THE e have seen how the oneway ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationTesting Hypotheses using SPSS
Is the mean hourly rate of male workers $2.00? TTest OneSample Statistics Std. Error N Mean Std. Deviation Mean 2997 2.0522 6.6282.2 OneSample Test Test Value = 2 95% Confidence Interval Mean of the
More informationUnit 29 ChiSquare GoodnessofFit Test
Unit 29 ChiSquare GoodnessofFit Test Objectives: To perform the chisquare hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni
More informationIndependent t Test (Comparing Two Means)
Independent t Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent ttest when to use the independent ttest the use of SPSS to complete an independent
More informationACTM State ExamStatistics
ACTM State ExamStatistics For the 25 multiplechoice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tiebreaker
More informationE205 Final: Version B
Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random
More informationChapter 16 Multiple Choice Questions (The answers are provided after the last question.)
Chapter 16 Multiple Choice Questions (The answers are provided after the last question.) 1. Which of the following symbols represents a population parameter? a. SD b. σ c. r d. 0 2. If you drew all possible
More informationHYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION
HYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate
More informationEPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM
EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable
More informationOneWay Analysis of Variance: A Guide to Testing Differences Between Multiple Groups
OneWay Analysis of Variance: A Guide to Testing Differences Between Multiple Groups In analysis of variance, the main research question is whether the sample means are from different populations. The
More informationSPSS on two independent samples. Two sample test with proportions. Paired ttest (with more SPSS)
SPSS on two independent samples. Two sample test with proportions. Paired ttest (with more SPSS) State of the course address: The Final exam is Aug 9, 3:30pm 6:30pm in B9201 in the Burnaby Campus. (One
More informationAnalysis of numerical data S4
Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 3rd November 2015 1/42 Hypothesis tests: numerical and ordinal data 1 group:
More informationOutline. Topic 4  Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares
Topic 4  Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test  Fall 2013 R 2 and the coefficient of correlation
More informationc. The factor is the type of TV program that was watched. The treatment is the embedded commercials in the TV programs.
STAT E150  Statistical Methods Assignment 9 Solutions Exercises 12.8, 12.13, 12.75 For each test: Include appropriate graphs to see that the conditions are met. Use Tukey's Honestly Significant Difference
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationAdditional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm
Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm
More informationStatistics II Final Exam  January Use the University stationery to give your answers to the following questions.
Statistics II Final Exam  January 2012 Use the University stationery to give your answers to the following questions. Do not forget to write down your name and class group in each page. Indicate clearly
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationGeneral Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test
Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics
More informationHypothesis Testing & Data Analysis. Statistics. Descriptive Statistics. What is the difference between descriptive and inferential statistics?
2 Hypothesis Testing & Data Analysis 5 What is the difference between descriptive and inferential statistics? Statistics 8 Tools to help us understand our data. Makes a complicated mess simple to understand.
More informationIntroduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 91
Introduction to Hypothesis Testing 91 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type
More informationOneWay Analysis of Variance (ANOVA) Example Problem
OneWay Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesistesting technique used to test the equality of two or more population (or treatment) means
More informationRegression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology
Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of
More informationtaken together, can provide strong support. Using a method for combining probabilities, it can be determined that combining the probability values of
taken together, can provide strong support. Using a method for combining probabilities, it can be determined that combining the probability values of 0.11 and 0.07 results in a probability value of 0.045.
More informationProvide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test.
Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations.
More informationTesting Group Differences using Ttests, ANOVA, and Nonparametric Measures
Testing Group Differences using Ttests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 354870348 Phone:
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationTwosample hypothesis testing, II 9.07 3/16/2004
Twosample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For twosample tests of the difference in mean, things get a little confusing, here,
More informationAnalysis of Variance ANOVA
Analysis of Variance ANOVA Overview We ve used the t test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.
More informationStatistics Review PSY379
Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationAbout Single Factor ANOVAs
About Single Factor ANOVAs TABLE OF CONTENTS About Single Factor ANOVAs... 1 What is a SINGLE FACTOR ANOVA... 1 Single Factor ANOVA... 1 Calculating Single Factor ANOVAs... 2 STEP 1: State the hypotheses...
More informationStatistics 112 Regression Cheatsheet Section 1B  Ryan Rosario
Statistics 112 Regression Cheatsheet Section 1B  Ryan Rosario I have found that the best way to practice regression is by brute force That is, given nothing but a dataset and your mind, compute everything
More informationStatistics for Management IISTAT 362Final Review
Statistics for Management IISTAT 362Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to
More informationCHAPTER 13. Experimental Design and Analysis of Variance
CHAPTER 13 Experimental Design and Analysis of Variance CONTENTS STATISTICS IN PRACTICE: BURKE MARKETING SERVICES, INC. 13.1 AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE Data Collection
More informationHypothesis testing S2
Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to
More informationStatistics and research
Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,
More informationStatistiek II. John Nerbonne. March 24, 2010. Information Science, Groningen Slides improved a lot by Harmut Fitz, Groningen!
Information Science, Groningen j.nerbonne@rug.nl Slides improved a lot by Harmut Fitz, Groningen! March 24, 2010 Correlation and regression We often wish to compare two different variables Examples: compare
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationExamining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish
Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish Statistics Statistics are quantitative methods of describing, analysing, and drawing inferences (conclusions)
More informationInferences About Differences Between Means Edpsy 580
Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at UrbanaChampaign Inferences About Differences Between Means Slide
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More informationIntroduction to Analysis of Variance (ANOVA) Limitations of the ttest
Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One Way ANOVA Limitations of the ttest Although the ttest is commonly used, it has limitations Can only
More information12: Analysis of Variance. Introduction
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
More informationPsychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!
Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on
More informationLinear Models in STATA and ANOVA
Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 42 A Note on NonLinear Relationships 44 Multiple Linear Regression 45 Removal of Variables 48 Independent Samples
More informationFactor B: Curriculum New Math Control Curriculum (B (B 1 ) Overall Mean (marginal) Females (A 1 ) Factor A: Gender Males (A 2) X 21
1 Factorial ANOVA The ANOVA designs we have dealt with up to this point, known as simple ANOVA or oneway ANOVA, had only one independent grouping variable or factor. However, oftentimes a researcher has
More informationUNDERSTANDING ANALYSIS OF COVARIANCE (ANCOVA)
UNDERSTANDING ANALYSIS OF COVARIANCE () In general, research is conducted for the purpose of explaining the effects of the independent variable on the dependent variable, and the purpose of research design
More informationChapter 7. Oneway ANOVA
Chapter 7 Oneway ANOVA Oneway ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The ttest of Chapter 6 looks
More informationAMS7: WEEK 8. CLASS 1. Correlation Monday May 18th, 2015
AMS7: WEEK 8. CLASS 1 Correlation Monday May 18th, 2015 Type of Data and objectives of the analysis Paired sample data (Bivariate data) Determine whether there is an association between two variables This
More informationNonParametric Tests (I)
Lecture 5: NonParametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of DistributionFree Tests (ii) Median Test for Two Independent
More informationContrasts ask specific questions as opposed to the general ANOVA null vs. alternative
Chapter 13 Contrasts and Custom Hypotheses Contrasts ask specific questions as opposed to the general ANOVA null vs. alternative hypotheses. In a oneway ANOVA with a k level factor, the null hypothesis
More informatione = random error, assumed to be normally distributed with mean 0 and standard deviation σ
1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.
More information4.4. Further Analysis within ANOVA
4.4. Further Analysis within ANOVA 1) Estimation of the effects Fixed effects model: α i = µ i µ is estimated by a i = ( x i x) if H 0 : µ 1 = µ 2 = = µ k is rejected. Random effects model: If H 0 : σa
More informationChapter 11: Two Variable Regression Analysis
Department of Mathematics Izmir University of Economics Week 1415 20142015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationComparing Means in Two Populations
Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we
More informationData Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments  Introduction
Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments  Introduction
More informationSTA201TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance
Principles of Statistics STA201TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis
More informationCOMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.
277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies
More informationresearch/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other
1 Hypothesis Testing Richard S. Balkin, Ph.D., LPCS, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric
More informationVariables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.
The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide
More informationp1^ = 0.18 p2^ = 0.12 A) 0.150 B) 0.387 C) 0.300 D) 0.188 3) n 1 = 570 n 2 = 1992 x 1 = 143 x 2 = 550 A) 0.270 B) 0.541 C) 0.520 D) 0.
Practice for chapter 9 and 10 Disclaimer: the actual exam does not mirror this. This is meant for practicing questions only. The actual exam in not multiple choice. Find the number of successes x suggested
More informationUNDERSTANDING THE INDEPENDENTSAMPLES t TEST
UNDERSTANDING The independentsamples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationBA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394
BA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete
More informationFor example, enter the following data in three COLUMNS in a new View window.
Statistics with Statview  18 Paired ttest A paired ttest compares two groups of measurements when the data in the two groups are in some way paired between the groups (e.g., before and after on the
More informationDDBA 8438: The t Test for Independent Samples Video Podcast Transcript
DDBA 8438: The t Test for Independent Samples Video Podcast Transcript JENNIFER ANN MORROW: Welcome to The t Test for Independent Samples. My name is Dr. Jennifer Ann Morrow. In today's demonstration,
More informationMultiple Linear Regression
Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is
More informationThe scatterplot indicates a positive linear relationship between waist size and body fat percentage:
STAT E150 Statistical Methods Multiple Regression Three percent of a man's body is essential fat, which is necessary for a healthy body. However, too much body fat can be dangerous. For men between the
More informationReporting Statistics in Psychology
This document contains general guidelines for the reporting of statistics in psychology research. The details of statistical reporting vary slightly among different areas of science and also among different
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationStatistical Models in R
Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 16233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova
More informationContents 1. Contents
Contents 1 Contents 3 Ksample Methods 2 3.1 Setup............................ 2 3.2 Classic Method Based on Normality Assumption..... 3 3.3 Permutation F test.................... 5 3.4 KruskalWallis
More informationBasic Statistcs Formula Sheet
Basic Statistcs Formula Sheet Steven W. ydick May 5, 0 This document is only intended to review basic concepts/formulas from an introduction to statistics course. Only meanbased procedures are reviewed,
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationReview on Univariate Analysis of Variance (ANOVA) Pekka Malo 30E00500 Quantitative Empirical Research Spring 2016
Review on Univariate Analysis of Variance (ANOVA) Pekka Malo 30E00500 Quantitative Empirical Research Spring 2016 Analysis of Variance Analysis of Variance (ANOVA) ~ special case of multiple regression,
More informationThe Dummy s Guide to Data Analysis Using SPSS
The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests
More informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More informationHow to choose a statistical test. Francisco J. Candido dos Reis DGOFMRP University of São Paulo
How to choose a statistical test Francisco J. Candido dos Reis DGOFMRP University of São Paulo Choosing the right test One of the most common queries in stats support is Which analysis should I use There
More information