ANOVA. February 12, 2015

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "ANOVA. February 12, 2015"

Transcription

1 ANOVA February 12, ANOVA models Last time, we discussed the use of categorical variables in multivariate regression. Often, these are encoded as indicator columns in the design matrix. In [1]: %%R url = salary.table = read.table(url, header=t) salary.table$e = factor(salary.table$e) salary.table$m = factor(salary.table$m) salary.lm = lm(s ~ X + E + M, salary.table) head(model.matrix(salary.lm)) (Intercept) X E2 E3 M Often, especially in experimental settings, we record only categorical variables. Such models are often referred to ANOVA (Analysis of Variance) models. These are generalizations of our favorite example, the two sample t-test. 1.1 Example: recovery time Suppose we want to understand the relationship between recovery time after surgery based on an patient s prior fitness. We group patients into three fitness levels: below average, average, above average. If you are in better shape before surgery, does it take less time to recover? In [2]: %%R url = rehab.table = read.table(url, header=t, sep=, ) rehab.table$fitness <- factor(rehab.table$fitness) head(rehab.table) 1

2 Fitness Time In [3]: %%R -h 800 -w 800 attach(rehab.table) boxplot(time ~ Fitness, col=c( red, green, blue )) 2

3 1.2 One-way ANOVA First generalization of two sample t-test: more than two groups. Observations are broken up into r groups with n i, 1 i r observations per group. Model: Y ij = µ + α i + ε ij, ε ij N(0, σ 2 ). Constraint: r i=1 α i = 0. This constraint is needed for identifiability. This is equivalent to only adding r 1 columns to the design matrix for this qualitative variable. This is not the same parameterization we get when only adding r columns, but it gives the same model. The estimates of α can be obtained from the estimates of β using R s default parameters. For a more detailed exploration into R s creation of design matrices, try reading the following tutorial on design matrices. 1.3 Remember, it s still a model (i.e. a plane) 1.4 Fitting the model Model is easy to fit: Ŷ ij = 1 n i n i j=1 Y ij = Y i. If observation is in i-th group: predicted mean is just the sample mean of observations in i-th group. Simplest question: is there any group (main) effect? H 0 : α 1 = = α r = 0? Test is based on F -test with full model vs. reduced model. Reduced model just has an intercept. Other questions: is the effect the same in groups 1 and 2? In [4]: %%R rehab.lm <- lm(time ~ Fitness) summary(rehab.lm) H 0 : α 1 = α 2? Call: lm(formula = Time ~ Fitness) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** Fitness ** Fitness e-06 *** 3

4 Signif. codes: 0 *** ** 0.01 * Residual standard error: on 21 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 2 and 21 DF, p-value: 4.129e-05 In [5]: %%R print(predict(rehab.lm, list(fitness=factor(c(1,2,3))))) c(mean(time[fitness == 1]), mean(time[fitness == 2]), mean(time[fitness == 3])) [1] Recall that the rows of the Coefficients table above do not correspond to the α parameter. For one thing, we would see three α s and their sum would have to be equal to 0. Also, the design matrix is the indicator coding we saw last time. In [6]: %%R head(model.matrix(rehab.lm)) (Intercept) Fitness2 Fitness There are ways to get different design matrices by using the contrasts argument. This is a bit above our pay grade at the moment. Upon inspection of the design matrix above, we see that the (Intercept) coefficient corresponds to the mean in Fitness==1, while Fitness==2 coefficient corresponds to the difference between the groups Fitness==2 and Fitness== ANOVA table Much of the information in an ANOVA model is contained in the ANOVA table. In [8]: make_table(anova_oneway) apply_theme( basic ) Out[8]: <ipy table.ipytable at 0x107d8c250> In [9]: %%R anova(rehab.lm) 4

5 Analysis of Variance Table Response: Time Df Sum Sq Mean Sq F value Pr(>F) Fitness e-05 *** Residuals Signif. codes: 0 *** ** 0.01 * Note that MST R measures variability of the cell means. If there is a group effect we expect this to be large relative to MSE. We see that under H 0 : α 1 = = α r = 0, the expected value of MST R and MSE is σ 2. This tells us how to test H 0 using ratio of mean squares, i.e. an F test. 1.6 Testing for any main effect Rows in the ANOVA table are, in general, independent. Therefore, under H 0 F = MST R MSE SST R = df T R SSE df E F dft R,df E the degrees of freedom come from the df column in previous table. Reject H 0 at level α if F > F 1 α,dft R,df E. In [10]: %%R F = / pval = 1 - pf(f, 2, 21) print(data.frame(f,pval)) F pval e Inference for linear combinations Suppose we want to infer something about r a i µ i where µ i = µ + α i is the mean in the i-th group. For example: H 0 : µ 1 µ 2 = 0 (same as H 0 : α 1 α 2 = 0)? i=1 For example: Is there a difference between below average and average groups in terms of rehab time? 5

6 We need to know ( r r Var a i Y i ) = σ 2 a 2 i. n i After this, the usual confidence intervals and t-tests apply. In [11]: %%R head(model.matrix(rehab.lm)) (Intercept) Fitness2 Fitness i=1 i=1 This means that the coefficient Fitness2 is the estimated difference between the two groups. In [12]: %%R detach(rehab.table) 1.8 Two-way ANOVA Often, we will have more than one variable we are changing Example After kidney failure, we suppose that the time of stay in hospital depends on weight gain between treatments and duration of treatment. We will model the log number of days as a function of the other two factors. In [14]: make_table(desc) apply_theme( basic ) Out[14]: <ipy table.ipytable at 0x107d8cd90> In [15]: %%R url = kidney.table = read.table(url, header=t) kidney.table$d = factor(kidney.table$duration) kidney.table$w = factor(kidney.table$weight) kidney.table$logdays = log(kidney.table$days + 1) attach(kidney.table) head(kidney.table) Days Duration Weight ID D W logdays

7 1.8.2 Two-way ANOVA model Second generalization of t-test: more than one grouping variable. Two-way ANOVA model: r groups in first factor m groups in second factor n ij in each combination of factor variables. Model: Y ijk = µ + α i + β j + (αβ) ij + ε ijk, ε ijk N(0, σ 2 ). In kidney example, r = 3 (weight gain), m = 2 (duration of treatment), n ij = 10 for all (i, j) Questions of interest Two-way ANOVA: main questions of interest Are there main effects for the grouping variables? Are there interaction effects: Interactions between factors H 0 : α 1 = = α r = 0, H 0 : β 1 = = β m = 0. H 0 : (αβ) ij = 0, 1 i r, 1 j m. We ve already seen these interactions in the IT salary example. An additive model says that the effects of the two factors occur additively such a model has no interactions. An interaction is present whenever the additive model does not hold Interaction plot In [16]: %%R -h 800 -w 800 interaction.plot(w, D, logdays, type= b, col=c( red, blue ), lwd=2, pch=c(23,24)) 7

8 When these broken lines are not parallel, there is evidence of an interaction. The one thing missing from this plot are errorbars. The above broken lines are clearly not parallel but there is measurement error. If the error bars were large then we might consider there to be no interaction, otherwise we might Parameterization Many constraints are needed, again for identifiability. Let s not worry too much about the details Constraints: r i=1 α i = 0 m j=1 β j = 0 m j=1 (αβ) ij = 0, 1 i r r i=1 (αβ) ij = 0, 1 j m. We should convince ourselves that we know have exactly r m free parameters. 8

9 1.8.7 Fitting the model Easy to fit when n ij = n (balanced) Ŷ ijk = Y ij = 1 n n Y ijk. k=1 Inference for combinations r m Var a ij Y ij = σ2 n i=1 j=1 r m i=1 j=1 a 2 ij. Usual t-tests, confidence intervals. In [17]: %%R kidney.lm = lm(logdays ~ D*W) summary(kidney.lm) Call: lm(formula = logdays ~ D * W) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-05 *** D W * W e-05 *** D2:W D2:W Signif. codes: 0 *** ** 0.01 * Residual standard error: on 54 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 5 and 54 DF, p-value: 2.301e Example Suppose we are interested in comparing the mean in (D = 1, W = 3) and (D = 2, W = 2) groups. The difference is E(Ȳ13 Ȳ22 ) By independence, its variance is Var(Ȳ13 ) + Var(Ȳ22 ) = 2σ2 n. 9

10 In [18]: %%R estimates = predict(kidney.lm, list(d=factor(c(1,2)), W=factor(c(3,2)))) print(estimates) sigma.hat = # from table above n = 10 # ten observations per group fit = estimates[1] - estimates[2] upper = fit + qt(0.975, 54) * sqrt(2 * sigma.hat^2 / n) lower = fit - qt(0.975,54) * sqrt(2 * sigma.hat^2 / n) data.frame(fit,lower,upper) fit lower upper In [19]: %%R head(model.matrix(kidney.lm)) (Intercept) D2 W2 W3 D2:W2 D2:W Finding predicted values The most direct way to compute predicted values is using the predict function In [20]: %%R predict(kidney.lm, list(d=factor(1),w=factor(1)), interval= confidence ) fit lwr upr ANOVA table In the balanced case, everything can again be summarized from the ANOVA table In [22]: make_table(anova_twoway) apply_theme( basic ) Out[22]: <ipy table.ipytable at 0x107d8c890> Tests using the ANOVA table Rows of the ANOVA table can be used to test various of the hypotheses we started out with. For instance, we see that under H 0 : (αβ) ij = 0, i, j the expected value of SSAB and SSE is σ 2 use these for an F -test testing for an interaction. 10

11 Under H 0 In [23]: %%R anova(kidney.lm) Analysis of Variance Table (m 1)(r 1) F = MSAB SSAB MSE = SSE (n 1)mr F (m 1)(r 1),(n 1)mr Response: logdays Df Sum Sq Mean Sq F value Pr(>F) D * W e-06 *** D:W Residuals Signif. codes: 0 *** ** 0.01 * We can also test for interactions using our usual approach In [24]: %%R anova(lm(logdays ~ D + W, kidney.table), kidney.lm) Analysis of Variance Table Model 1: logdays ~ D + W Model 2: logdays ~ D * W Res.Df RSS Df Sum of Sq F Pr(>F) Some caveats about R formulae While we see that it is straightforward to form the interactions test using our usual anova function approach, we generally cannot test for main effects by this approach. In [25]: %%R lm_no_main_weight = lm(logdays ~ D + W:D) anova(lm_no_main_weight, kidney.lm) Analysis of Variance Table Model 1: logdays ~ D + W:D Model 2: logdays ~ D * W Res.Df RSS Df Sum of Sq F Pr(>F) e-15 In fact, these models are identical in terms of their planes or their fitted values. What has happened is that R has formed a different design matrix using its rules for formula objects. 11

12 In [26]: %%R lm1 = lm(logdays ~ D + W:D) lm2 = lm(logdays ~ D + W:D + W) anova(lm1, lm2) Analysis of Variance Table Model 1: logdays ~ D + W:D Model 2: logdays ~ D + W:D + W Res.Df RSS Df Sum of Sq F Pr(>F) e ANOVA tables in general So far, we have used anova to compare two models. In this section, we produced tables for just 1 model. This also works for any regression model, though we have to be a little careful about interpretation. Let s revisit the job aptitude test data from last section. In [27]: %%R url = jobtest.table <- read.table(url, header=t) jobtest.table$ethn <- factor(jobtest.table$ethn) jobtest.lm = lm(jperf ~ TEST * ETHN, jobtest.table) summary(jobtest.lm) Call: lm(formula = JPERF ~ TEST * ETHN, data = jobtest.table) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) TEST ETHN TEST:ETHN Signif. codes: 0 *** ** 0.01 * Residual standard error: on 16 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 3 and 16 DF, p-value: Now, let s look at the anova output. We ll see the results don t match. In [28]: %%R anova(jobtest.lm) 12

13 Analysis of Variance Table Response: JPERF Df Sum Sq Mean Sq F value Pr(>F) TEST *** ETHN TEST:ETHN Residuals Signif. codes: 0 *** ** 0.01 * The difference is how the Sum Sq columns is created. In the anova output, terms in the response are added sequentially. We can see this by comparing these two models directly. The F statistic doesn t agree because the MSE above is computed in the fullest model, but the Sum of Sq is correct. In [29]: %%R anova(lm(jperf ~ TEST, jobtest.table), lm(jperf ~ TEST + ETHN, jobtest.table)) Analysis of Variance Table Model 1: JPERF ~ TEST Model 2: JPERF ~ TEST + ETHN Res.Df RSS Df Sum of Sq F Pr(>F) Similarly, the first Sum Sq in anova can be found by: In [30]: %%R anova(lm(jperf ~ 1, jobtest.table), lm(jperf ~ TEST, jobtest.table)) Analysis of Variance Table Model 1: JPERF ~ 1 Model 2: JPERF ~ TEST Res.Df RSS Df Sum of Sq F Pr(>F) *** Signif. codes: 0 *** ** 0.01 * There are ways to produce an ANOVA table whose p-values agree with summary. This is done by an ANOVA table that uses Type-III sum of squares. In [31]: %%R library(car) Anova(jobtest.lm, type=3) 13

14 Anova Table (Type III tests) Response: JPERF Sum Sq Df F value Pr(>F) (Intercept) TEST ETHN TEST:ETHN Residuals Signif. codes: 0 *** ** 0.01 * In [32]: %%R summary(jobtest.lm) Call: lm(formula = JPERF ~ TEST * ETHN, data = jobtest.table) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) TEST ETHN TEST:ETHN Signif. codes: 0 *** ** 0.01 * Residual standard error: on 16 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 3 and 16 DF, p-value: Fixed and random effects In kidney & rehab examples, the categorical variables are well-defined categories: below average fitness, long duration, etc. In some designs, the categorical variable is subject. Simplest example: repeated measures, where more than one (identical) measurement is taken on the same individual. In this case, the group effect α i is best thought of as random because we only sample a subset of the entire population. 14

15 2.0.1 When to use random effects? A group effect is random if we can think of the levels we observe in that group to be samples from a larger population. Example: if collecting data from different medical centers, center might be thought of as random. Example: if surveying students on different campuses, campus may be a random effect Example: sodium content in beer How much sodium is there in North American beer? How much does this vary by brand? Observations: for 6 brands of beer, we recorded the sodium content of 8 12 ounce bottles. Questions of interest: what is the grand mean sodium content? How much variability is there from brand to brand? Individuals in this case are brands, repeated measures are the 8 bottles. In [33]: %%R url = sodium.table = read.table(url, header=t) sodium.table$brand = factor(sodium.table$brand) sodium.lm = lm(sodium ~ brand, sodium.table) anova(sodium.lm) Analysis of Variance Table Response: sodium Df Sum Sq Mean Sq F value Pr(>F) brand < 2.2e-16 *** Residuals Signif. codes: 0 *** ** 0.01 * One-way random effects model Assuming that cell-sizes are the same, i.e. equal observations for each subject (brand of beer). Observations Y ij µ + α i + ε ij, 1 i r, 1 j n ε ij N(0, σ 2 ɛ ), 1 i r, 1 j n α i N(0, σ 2 α), 1 i r. Parameters: µ is the population mean; σ 2 ɛ is the measurement variance (i.e. how variable are the readings from the machine that reads the sodium content?); σ 2 α is the population variance (i.e. how variable is the sodium content of beer across brands). 15

16 2.0.4 Modelling the variance In random effects model, the observations are no longer independent (even if ε s are independent Cov(Y ij, Y i j ) = ( σ 2 α + σ 2 ɛ δ j,j ) δi,i. In more complicated models, this makes maximum likelihood estimation more complicated: least squares is no longer the best solution. It s no longer a plane! This model has a very simple model for the mean, it just has a slightly more complex model for the variance. Shortly we ll see other more complex models of the variance: Weighted Least Squares Correlated Errors Fitting the model The MLE (Maximum Likelihood Estimator) is found by minimizing 2 log l(µ, σ 2 ɛ, σ 2 α Y ) = r [ (Y i µ) T (σɛ 2 I ni n i + σα11 2 T ) 1 (Y i µ) i=1 + log ( det(σ 2 ɛ I ni n i + σ 2 α11 T ) )]. THe function l(µ, σ 2 ɛ, σ 2 α) is called the likelihood function Fitting the model in balanced design Only one parameter in the mean function µ. - When cell sizes are the same (balanced), Unbalanced models: use numerical optimizer. µ = Y = 1 Y ij. nr This also changes estimates of σ 2 ɛ see ANOVA table. We might guess that df = nr 1 and This is not correct. σ 2 = 1 nr 1 i,j (Y ij Y ) 2. In [34]: %%R library(nlme) sodium.lme = lme(fixed=sodium~1,random=~1 brand, data=sodium.table) summary(sodium.lme) Linear mixed-effects model fit by REML Data: sodium.table AIC BIC loglik Random effects: i,j 16

17 Formula: ~1 brand (Intercept) Residual StdDev: Fixed effects: sodium ~ 1 Value Std.Error DF t-value p-value (Intercept) Standardized Within-Group Residuals: Min Q1 Med Q3 Max Number of Observations: 48 Number of Groups: 6 For reasons I m not sure of, the degrees of freedom don t agree with our ANOVA, though we do find the correct SE for our estimate of µ: In [35]: %%R MSTR = anova(sodium.lm)$mean[1] sqrt(mstr/48) [1] The intervals formed by lme use the 42 degrees of freedom, but are otherwise the same: In [36]: %%R intervals(sodium.lme) Approximate 95% confidence intervals Fixed effects: lower est. upper (Intercept) attr(,"label") [1] "Fixed effects:" Random Effects: Level: brand lower est. upper sd((intercept)) Within-group standard error: lower est. upper In [37]: %%R center = mean(sodium.table$sodium) lwr = center - sqrt(mstr / 48) * qt(0.975,42) upr = center + sqrt(mstr / 48) * qt(0.975,42) data.frame(lwr, center, upr) 17

18 lwr center upr Using our degrees of freedom as 7 yields slightly wider intervals In [38]: %%R center = mean(sodium.table$sodium) lwr = center - sqrt(mstr / 48) * qt(0.975,7) upr = center + sqrt(mstr / 48) * qt(0.975,7) data.frame(lwr, center, upr) lwr center upr ANOVA table Again, the information needed can be summarized in an ANOVA table. In [40]: make_table(anova_oneway) apply_theme( basic ) Out[40]: <ipy table.ipytable at 0x107d8c990> ANOVA table is still useful to setup tests: the same F statistics for fixed or random will work here. Test for random effect: H 0 : σ 2 α = 0 based on Inference for µ F = MST R MSE F r 1,(n 1)r under H 0. Easy to check that E(Y ) = µ Var(Y ) = σ2 ɛ + nσα 2. rn To come up with a t statistic that we can use for test, CIs, we need to find an estimate of Var(Y ). ANOVA table says E(MST R) = nσ 2 α + σ 2 ɛ which suggests Degrees of freedom Why r 1 degrees of freedom? Y µ MST R rn t r 1. Imagine we could record an infinite number of observations for each individu al, so that Y i µ + α i. To learn anything about µ we still only have r observations (µ 1,..., µ r ). Sampling more within an individual cannot narrow the CI for µ. 18

19 Estimating σ 2 α We have seen estimates of µ and σ 2 ɛ. Only one parameter remains. Based on the ANOVA table, we see that σα 2 = 1 (E(MST R) E(MSE)). n This suggests the estimate ˆσ 2 µ = 1 (MST R MSE). n However, this estimate can be negative! Many such computational difficulties arise in random (and mixed) effects models. 2.1 Mixed effects model The one-way random effects ANOVA is a special case of a so-called mixed effects model: Y n 1 = X n p β p 1 + Z n q γ q 1 γ N(0, Σ). Various models also consider restrictions on Σ (e.g. diagonal, unrestricted, block diagonal, etc.) Our multiple linear regression model is a (very simple) mixed-effects model with q = n, Z = I n n Σ = σ 2 I n n. 19

Fixed vs. Random Effects

Fixed vs. Random Effects Statistics 203: Introduction to Regression and Analysis of Variance Fixed vs. Random Effects Jonathan Taylor - p. 1/19 Today s class Implications for Random effects. One-way random effects ANOVA. Two-way

More information

Multiple Linear Regression

Multiple 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 information

E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F

E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F Random and Mixed Effects Models (Ch. 10) Random effects models are very useful when the observations are sampled in a highly structured way. The basic idea is that the error associated with any linear,

More information

Regression in ANOVA. James H. Steiger. Department of Psychology and Human Development Vanderbilt University

Regression in ANOVA. James H. Steiger. Department of Psychology and Human Development Vanderbilt University Regression in ANOVA James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 30 Regression in ANOVA 1 Introduction 2 Basic Linear

More information

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

More information

We extended the additive model in two variables to the interaction model by adding a third term to the equation.

We extended the additive model in two variables to the interaction model by adding a third term to the equation. Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic

More information

Statistical Models in R

Statistical Models in R Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

More information

Recall this chart that showed how most of our course would be organized:

Recall this chart that showed how most of our course would be organized: Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

Statistical Models in R

Statistical Models in R Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova

More information

Comparing Nested Models

Comparing Nested Models Comparing Nested Models ST 430/514 Two models are nested if one model contains all the terms of the other, and at least one additional term. The larger model is the complete (or full) model, and the smaller

More information

Stat 411/511 ANOVA & REGRESSION. Charlotte Wickham. stat511.cwick.co.nz. Nov 31st 2015

Stat 411/511 ANOVA & REGRESSION. Charlotte Wickham. stat511.cwick.co.nz. Nov 31st 2015 Stat 411/511 ANOVA & REGRESSION Nov 31st 2015 Charlotte Wickham stat511.cwick.co.nz This week Today: Lack of fit F-test Weds: Review email me topics, otherwise I ll go over some of last year s final exam

More information

One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

More information

N-Way Analysis of Variance

N-Way Analysis of Variance N-Way Analysis of Variance 1 Introduction A good example when to use a n-way ANOVA is for a factorial design. A factorial design is an efficient way to conduct an experiment. Each observation has data

More information

Classical Hypothesis Testing in R. R can do all of the common analyses that are available in SPSS, including:

Classical Hypothesis Testing in R. R can do all of the common analyses that are available in SPSS, including: Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including: Classical Hypothesis Testing in R R can do all of the common analyses that are available in

More information

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 > catch

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1. 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 1-propZint 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 information

Week 5: Multiple Linear Regression

Week 5: Multiple Linear Regression BUS41100 Applied Regression Analysis Week 5: Multiple Linear Regression Parameter estimation and inference, forecasting, diagnostics, dummy variables Robert B. Gramacy The University of Chicago Booth School

More information

Lecture 5 Hypothesis Testing in Multiple Linear Regression

Lecture 5 Hypothesis Testing in Multiple Linear Regression Lecture 5 Hypothesis Testing in Multiple Linear Regression BIOST 515 January 20, 2004 Types of tests 1 Overall test Test for addition of a single variable Test for addition of a group of variables Overall

More information

Generalized Linear Models

Generalized Linear Models Generalized Linear Models We have previously worked with regression models where the response variable is quantitative and normally distributed. Now we turn our attention to two types of models where the

More information

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters) An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

More information

12-1 Multiple Linear Regression Models

12-1 Multiple Linear Regression Models 12-1.1 Introduction Many applications of regression analysis involve situations in which there are more than one regressor variable. A regression model that contains more than one regressor variable is

More information

Regression, least squares

Regression, least squares Regression, least squares Joe Felsenstein Department of Genome Sciences and Department of Biology Regression, least squares p.1/24 Fitting a straight line X Two distinct cases: The X values are chosen

More information

Chapter 7. One-way ANOVA

Chapter 7. One-way ANOVA Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

More information

Introducing the Multilevel Model for Change

Introducing the Multilevel Model for Change Department of Psychology and Human Development Vanderbilt University GCM, 2010 1 Multilevel Modeling - A Brief Introduction 2 3 4 5 Introduction In this lecture, we introduce the multilevel model for change.

More information

Psychology 205: Research Methods in Psychology

Psychology 205: Research Methods in Psychology Psychology 205: Research Methods in Psychology Using R to analyze the data for study 2 Department of Psychology Northwestern University Evanston, Illinois USA November, 2012 1 / 38 Outline 1 Getting ready

More information

Statistics II Final Exam - January Use the University stationery to give your answers to the following questions.

Statistics 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 information

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a

More information

EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION

EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 5-10 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day

More information

Using R for Linear Regression

Using R for Linear Regression Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional

More information

Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares

Outline. 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 information

5. Linear Regression

5. Linear Regression 5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

More information

Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

More information

An analysis method for a quantitative outcome and two categorical explanatory variables.

An analysis method for a quantitative outcome and two categorical explanatory variables. Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

More information

Time-Series Regression and Generalized Least Squares in R

Time-Series Regression and Generalized Least Squares in R Time-Series Regression and Generalized Least Squares in R An Appendix to An R Companion to Applied Regression, Second Edition John Fox & Sanford Weisberg last revision: 11 November 2010 Abstract Generalized

More information

And sample sizes > tapply(count, spray, length) A B C D E F And a boxplot: > boxplot(count ~ spray) How does the data look?

And sample sizes > tapply(count, spray, length) A B C D E F And a boxplot: > boxplot(count ~ spray) How does the data look? ANOVA in R 1-Way ANOVA We re going to use a data set called InsectSprays. 6 different insect sprays (1 Independent Variable with 6 levels) were tested to see if there was a difference in the number of

More information

Random effects and nested models with SAS

Random effects and nested models with SAS Random effects and nested models with SAS /************* classical2.sas ********************* Three levels of factor A, four levels of B Both fixed Both random A fixed, B random B nested within A ***************************************************/

More information

MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL. by Michael L. Orlov Chemistry Department, Oregon State University (1996)

MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL. by Michael L. Orlov Chemistry Department, Oregon State University (1996) MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL by Michael L. Orlov Chemistry Department, Oregon State University (1996) INTRODUCTION In modern science, regression analysis is a necessary part

More information

SCHOOL OF MATHEMATICS AND STATISTICS

SCHOOL OF MATHEMATICS AND STATISTICS RESTRICTED OPEN BOOK EXAMINATION (Not to be removed from the examination hall) Data provided: Statistics Tables by H.R. Neave MAS5052 SCHOOL OF MATHEMATICS AND STATISTICS Basic Statistics Spring Semester

More information

Lets suppose we rolled a six-sided die 150 times and recorded the number of times each outcome (1-6) occured. The data is

Lets suppose we rolled a six-sided die 150 times and recorded the number of times each outcome (1-6) occured. The data is In this lab we will look at how R can eliminate most of the annoying calculations involved in (a) using Chi-Squared tests to check for homogeneity in two-way tables of catagorical data and (b) computing

More information

Statistics Review PSY379

Statistics 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 information

Paired Differences and Regression

Paired Differences and Regression Paired Differences and Regression Students sometimes have difficulty distinguishing between paired data and independent samples when comparing two means. One can return to this topic after covering simple

More information

Chapter 11: Two Variable Regression Analysis

Chapter 11: Two Variable Regression Analysis Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

More information

0.1 Multiple Regression Models

0.1 Multiple Regression Models 0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different

More information

Lecture 11: Confidence intervals and model comparison for linear regression; analysis of variance

Lecture 11: Confidence intervals and model comparison for linear regression; analysis of variance Lecture 11: Confidence intervals and model comparison for linear regression; analysis of variance 14 November 2007 1 Confidence intervals and hypothesis testing for linear regression Just as there was

More information

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association

More information

MIXED MODEL ANALYSIS USING R

MIXED MODEL ANALYSIS USING R Research Methods Group MIXED MODEL ANALYSIS USING R Using Case Study 4 from the BIOMETRICS & RESEARCH METHODS TEACHING RESOURCE BY Stephen Mbunzi & Sonal Nagda www.ilri.org/rmg www.worldagroforestrycentre.org/rmg

More information

Introduction to Stata

Introduction to Stata Introduction to Stata September 23, 2014 Stata is one of a few statistical analysis programs that social scientists use. Stata is in the mid-range of how easy it is to use. Other options include SPSS,

More information

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups One-Way 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 information

where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.

where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis. Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Multiple Linear Regression. Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables.

Multiple Linear Regression. Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables. 1 Multiple Linear Regression Basic Concepts Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables. In simple linear regression, we had

More information

Exercise Page 1 of 32

Exercise Page 1 of 32 Exercise 10.1 (a) Plot wages versus LOS. Describe the relationship. There is one woman with relatively high wages for her length of service. Circle this point and do not use it in the rest of this exercise.

More information

An Sweave Demo. Charles J. Geyer. July 27, latex

An Sweave Demo. Charles J. Geyer. July 27, latex An Sweave Demo Charles J. Geyer July 27, 2010 This is a demo for using the Sweave command in R. To get started make a regular L A TEX file (like this one) but give it the suffix.rnw instead of.tex and

More information

Testing for Lack of Fit

Testing for Lack of Fit Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

Simple Linear Regression

Simple 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 information

Statistics in Geophysics: Linear Regression II

Statistics in Geophysics: Linear Regression II Statistics in Geophysics: Linear Regression II Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/28 Model definition Suppose we have the following

More information

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 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 information

Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools. Tools for Summarizing Data Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

More information

ANOVA Designs - Part II. Nested Designs. Nested Designs. Nested Designs (NEST) Design Linear Model Computation

ANOVA Designs - Part II. Nested Designs. Nested Designs. Nested Designs (NEST) Design Linear Model Computation ANOVA Designs - Part II Nested Designs (NEST) Design Linear Model Computation Example NCSS s (FACT) Design Linear Model Computation Example NCSS RCB Factorial (Combinatorial Designs) Nested Designs A nested

More information

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients ( Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

More information

Statistics 112 Regression Cheatsheet Section 1B - Ryan Rosario

Statistics 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 information

Part II. Multiple Linear Regression

Part II. Multiple Linear Regression Part II Multiple Linear Regression 86 Chapter 7 Multiple Regression A multiple linear regression model is a linear model that describes how a y-variable relates to two or more xvariables (or transformations

More information

EXPECTED MEAN SQUARES AND MIXED MODEL ANALYSES. This will become more important later in the course when we discuss interactions.

EXPECTED MEAN SQUARES AND MIXED MODEL ANALYSES. This will become more important later in the course when we discuss interactions. EXPECTED MEN SQURES ND MIXED MODEL NLYSES Fixed vs. Random Effects The choice of labeling a factor as a fixed or random effect will affect how you will make the F-test. This will become more important

More information

Simple Linear Regression Inference

Simple 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 information

Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

More information

Math 141. Lecture 24: Model Comparisons and The F-test. Albyn Jones 1. 1 Library jones/courses/141

Math 141. Lecture 24: Model Comparisons and The F-test. Albyn Jones 1. 1 Library jones/courses/141 Math 141 Lecture 24: Model Comparisons and The F-test Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Nested Models Two linear models are Nested if one (the restricted

More information

Quantitative Understanding in Biology Module II: Model Parameter Estimation Lecture I: Linear Correlation and Regression

Quantitative Understanding in Biology Module II: Model Parameter Estimation Lecture I: Linear Correlation and Regression Quantitative Understanding in Biology Module II: Model Parameter Estimation Lecture I: Linear Correlation and Regression Correlation Linear correlation and linear regression are often confused, mostly

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

Regression step-by-step using Microsoft Excel

Regression step-by-step using Microsoft Excel Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression

More information

Two-Variable Regression: Interval Estimation and Hypothesis Testing

Two-Variable Regression: Interval Estimation and Hypothesis Testing Two-Variable Regression: Interval Estimation and Hypothesis Testing Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing

More information

(d) True or false? When the number of treatments a=9, the number of blocks b=10, and the other parameters r =10 and k=9, it is a BIBD design.

(d) True or false? When the number of treatments a=9, the number of blocks b=10, and the other parameters r =10 and k=9, it is a BIBD design. PhD Qualifying exam Methodology Jan 2014 Solutions 1. True or false question - only circle "true " or "false" (a) True or false? F-statistic can be used for checking the equality of two population variances

More information

Lecture Outline (week 13)

Lecture Outline (week 13) Lecture Outline (week 3) Analysis of Covariance in Randomized studies Mixed models: Randomized block models Repeated Measures models Pretest-posttest models Analysis of Covariance in Randomized studies

More information

Final Exam Practice Problem Answers

Final 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 information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

data visualization and regression

data visualization and regression data visualization and regression Sepal.Length 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 I. setosa I. versicolor I. virginica I. setosa I. versicolor I. virginica Species Species

More information

n + n log(2π) + n log(rss/n)

n + n log(2π) + n log(rss/n) There is a discrepancy in R output from the functions step, AIC, and BIC over how to compute the AIC. The discrepancy is not very important, because it involves a difference of a constant factor that cancels

More information

Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 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 One-way ANOVA To test the null hypothesis that several population means are equal,

More information

Experimental Designs (revisited)

Experimental Designs (revisited) Introduction to ANOVA Copyright 2000, 2011, J. Toby Mordkoff Probably, the best way to start thinking about ANOVA is in terms of factors with levels. (I say this because this is how they are described

More information

Regression 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 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 information

CS 147: Computer Systems Performance Analysis

CS 147: Computer Systems Performance Analysis CS 147: Computer Systems Performance Analysis One-Factor Experiments CS 147: Computer Systems Performance Analysis One-Factor Experiments 1 / 42 Overview Introduction Overview Overview Introduction Finding

More information

Analysis of Variance. MINITAB User s Guide 2 3-1

Analysis of Variance. MINITAB User s Guide 2 3-1 3 Analysis of Variance Analysis of Variance Overview, 3-2 One-Way Analysis of Variance, 3-5 Two-Way Analysis of Variance, 3-11 Analysis of Means, 3-13 Overview of Balanced ANOVA and GLM, 3-18 Balanced

More information

Simple Predictive Analytics Curtis Seare

Simple Predictive Analytics Curtis Seare Using Excel to Solve Business Problems: Simple Predictive Analytics Curtis Seare Copyright: Vault Analytics July 2010 Contents Section I: Background Information Why use Predictive Analytics? How to use

More information

The F distribution and the basic principle behind ANOVAs. Situating ANOVAs in the world of statistical tests

The F distribution and the basic principle behind ANOVAs. Situating ANOVAs in the world of statistical tests Tutorial The F distribution and the basic principle behind ANOVAs Bodo Winter 1 Updates: September 21, 2011; January 23, 2014; April 24, 2014; March 2, 2015 This tutorial focuses on understanding rather

More information

Multivariate Logistic Regression

Multivariate Logistic Regression 1 Multivariate Logistic Regression As in univariate logistic regression, let π(x) represent the probability of an event that depends on p covariates or independent variables. Then, using an inv.logit formulation

More information

ANALYSING LIKERT SCALE/TYPE DATA, ORDINAL LOGISTIC REGRESSION EXAMPLE IN R.

ANALYSING LIKERT SCALE/TYPE DATA, ORDINAL LOGISTIC REGRESSION EXAMPLE IN R. ANALYSING LIKERT SCALE/TYPE DATA, ORDINAL LOGISTIC REGRESSION EXAMPLE IN R. 1. Motivation. Likert items are used to measure respondents attitudes to a particular question or statement. One must recall

More information

The scatterplot indicates a positive linear relationship between waist size and body fat percentage:

The scatterplot indicates a positive linear relationship between waist size and body fat percentage: STAT E-150 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 information

Statistical Functions in Excel

Statistical Functions in Excel Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

More information

CHAPTER 13. Experimental Design and Analysis of Variance

CHAPTER 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 information

T O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

Section 13, Part 1 ANOVA. Analysis Of Variance

Section 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 information

MODEL I: DRINK REGRESSED ON GPA & MALE, WITHOUT CENTERING

MODEL I: DRINK REGRESSED ON GPA & MALE, WITHOUT CENTERING Interpreting Interaction Effects; Interaction Effects and Centering Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015 Models with interaction effects

More information

Linear Regression with One Regressor

Linear Regression with One Regressor Linear Regression with One Regressor Michael Ash Lecture 10 Analogy to the Mean True parameter µ Y β 0 and β 1 Meaning Central tendency Intercept and slope E(Y ) E(Y X ) = β 0 + β 1 X Data Y i (X i, Y

More information

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

More information

Introduction to Regression and Data Analysis

Introduction to Regression and Data Analysis Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

More information

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

e = 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 information

Didacticiel - Études de cas

Didacticiel - Études de cas 1 Topic Regression analysis with LazStats (OpenStat). LazStat 1 is a statistical software which is developed by Bill Miller, the father of OpenStat, a wellknow tool by statisticians since many years. These

More information

SAS Syntax and Output for Data Manipulation:

SAS Syntax and Output for Data Manipulation: Psyc 944 Example 5 page 1 Practice with Fixed and Random Effects of Time in Modeling Within-Person Change The models for this example come from Hoffman (in preparation) chapter 5. We will be examining

More information