Hypothesis Testing and Regression Analysis (PDF)

Course structure i. Foundations ii. The empirical research process 1. Research question, theory, and hypotheses 2. Research design 3. Sampling and measurement 4. Data collection 5. Hypotheses testing 6. Dissemination iii. Research ethics...

Course structure i. Foundations ii. The empirical research process 1. Research question, theory, and hypotheses 2. Research design 3. Sampling and measurement 4. Data collection 5. Hypotheses testing 6. Dissemination iii. Research ethics Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 1 Learning goals of this module  Learning goals – … understand the key idea of inferential statistics – … understand the process of hypothesis testing – … recap the primary theoretical statistical distributions – … learn to test hypotheses with regressions – … learn to interpret basic regression output – … understand the terms moderation and mediation in the context of regression analyses  Readings – Trochim, W., Donnelly, J. P., & Arora, K. (2020). Research Methods: The Essential Knowledge Base. Conjoint. ly: Pyrmont, Australia. Chapters 11 and 12 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 2  Technische Universität München i) Probability distributions Overview of probability distributions Probability distributions Discrete Continuous Binomial Normal Student t - Chi-squared or F-distribution distribution distribution distribution 2 -distribution Standard normal distribution Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 4 Normal distribution  Normal distribution: family of distributions with each of them being defined by the mean µ and the standard deviation σ – Symmetric distribution around the mean; bell-shaped – The spread of the curve depends on the standard deviation: larger spread in case of higher standard deviations 2.0 1.6 1.2 0.8 0.4 0.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 X μ = 0; σ² = 1.0 μ = 0; σ² = 0.2 μ = 0; σ² = 5.0 μ = -2; σ² = 0.5 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 5 Normal distribution  Standardized normal distribution: µ = 0 and σ = 1 – Area under the curve can be interpreted as probabilities X̄ = 0.3413 = 0.3413 = 0.1359 = 0.1359 -1.96σ 1.96σ 95% of values Z scores -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Standard Deviations from -4σ -3σ -2σ -1σ 0 +1σ +2σ +3σ +4σ the Mean Cumulative % 0.1% 2.3% 15.9% 50% 84.1% 97.7% 99.9% Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 6 Student distribution / t-Distribution  Student distribution: Symmetric distribution around the mean, median, and mode – Similar to normal distribution, but with wider tails – Form also dependent on the size of the sample, the smaller the sample the wider the tails: for every n a separate table is needed – Converges with a normal distribution for n > 29 t, df = 1 t, df = 2 t, df = 4 t, df = 8 normal Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 7  Technische Universität München See exercise session “Statistics recap”  Technische Universität München Hypothesis testing Example: Am I broke?  We formulate the research hypothesis as two hypotheses – H0: I am not broke (null hypothesis) – H1: I am broke (research hypothesis)  Determine critical value (C)  C ≤ 0 EUR (in the bank account)  Carry out the measurement (check bank account)  Bank = -125 EUR  Conclude  -125 EUR ≤ 0 EUR  Reject H0 and accept H1  I am broke Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 10 Foundations of statistical inference Parameter Population Sampling error: Inference difference between about the the parameters and population the statistics Estimate/statistic Sample  Sampling error arises any time a sample is used, thus there is always some degree of uncertainty in statistics  We can use inferential statistics to estimate these parameters in a way that takes sampling error into account  Inferential statistics: Techniques that allow us to use a sample to make generalizations about the population Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 11 Example: Is this population broke?  We formulate the research hypothesis as two hypotheses, whereby µ is the population parameter (here: average money in the bank account) – H0: µ > 0 (the population is not broke) – H1: µ ≤ 0 (the population is broke)  Sample: n = 30 individuals, 𝑥ҧ = -5 EUR, s = 25 EUR  Compute test statistic (here t-test) ҧ 𝑥−µ −5−0 t= n= 30 = −1.095 𝑠 25  Get critical value (t-distribution, ∝= 5%)  C = -1.699  Conclude  t = -1.095 > -1.699 = C  Do not reject H0  There is not enough evidence to claim that the population is broke at the significance level ∝ Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 12 Hypothesis testing: process Draw a Decide on a Select a suitable State H0 and H1 Calculate the Calculate the conclusion and significance statistical testing hypothesis critical value test statistic interpret the level α procedure results Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 13 1. State the null (H0) and alternative (H1) hypothesis  We formulate the research hypothesis as two hypotheses – The null hypothesis, H0: µ = 0 Status quo is assumed to be true: there is no difference Begin with the assumption that the null hypothesis is true – The alternative hypothesis, H1: µ  0 Is the opposite of the null hypothesis (i.e., challenges the null) This is the hypothesis that is expected by the researcher  Idea: Rejection of H0 leads to the acceptance of H1  These two hypotheses are mutually exclusive and exhaustive: either H0 or H1 is accepted Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 14 1. State the null (H0) and alternative (H1) hypothesis  Why is it all about rejection?  Consider we want to proof that all swans are white: we would have to check if every swan in the population is white  The rejection-approach is much easier: we only have to find one non-white swan to conclude – H0: all swans are white – H1: not all swans are white Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 15 1. State the null (H0) and alternative (H1) hypothesis  Hypothesis testing in action: criminal trials  In a criminal trial the hypotheses are: – H0: defendant is innocent – H1: defendant is guilty  H0 (innocent) is rejected if H1 (guilty) is supported by evidence beyond reasonable doubt  Keep in mind: Failure to reject H0 (prove guilty) does not imply innocence, only that the evidence is insufficient to reject it Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 16 1. State the null (H0) and alternative (H1) hypothesis  One-tailed (left or right-tailed) – Example left-tailed: In 1990, the average weight of people in the population was 73 kg. Hypothesis: Today, the average weight (weight) is lower than it was in 1990. H0: weight ≥ 73 kg H1: weight < 73 kg – Example right-tailed: In 1990, the average height of people in the population was 168 cm. Hypothesis: Today, the average height (height) is greater than it was in 1990. H0: height ≤ 168 cm H1: height > 168 cm  Two-tailed – Example 1: In 1990, the average income (income) of people in the population was 20,000 Euro. Hypothesis: Today, the average income is different than it was in 1990. H0: income = 20,000 Euro H1: income  20,000 Euro – Example 2: An experiment was carried out, in which the treatment group received a one-time payment and the control group received no payment. Hypothesis: Happiness (happiness) differs between the treatment (T) and control group (C). H0: happinessT = happinessC H1: happinessT  happinessC Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 17 2. Decide on a significance level α  When do we have enough evidence to reject the null hypothesis?  The confidence with which we reject or accept the null hypothesis depends on the significance level  In hypothesis testing, there are four possible outcomes Truth H0 is true H0 is not true False Do not Correct negative reject H0 decision Decision (Type II error) False positive Correct Reject H0 (Type I error) decision  The significance level α is the maximum probability of committing a type I error – For example, if α = 5% then there is a 5% chance of rejecting a true null hypothesis Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 18 2. Decide on a significance level α  Type I and Type II errors in a criminal trial Truth H0 is not true H0 is true (innocent) (guilty) Do not Criminal judged innocent reject H0 Correct decision (Type II error) (acquit) Decision Innocent person Reject H0 convicted Correct decision (send to jail) (Type I error) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 19 3. Select a suitable statistical testing procedure  (continued) When do we have enough evidence to reject the null hypothesis? 0.35 0.3 0.25 0.2 P(X=x) 0.15 0.1 0.05 0 -33 -22 -11 0 11 22 33 Money in the bank account (€) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 20 3. Select a suitable statistical testing procedure  Test statistic: value computed from the sample data that is used in making a decision about the rejection of the null hypothesis – The idea is to adjust for “noise” or the distribution – The choice of the test statistic depends on the statistic of interest and the type of data Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 21 3. Select a suitable statistical testing procedure Hypothesis tests univariate bivariate Tests for Tests for Tests for Tests for Tests for Tests for central central spread form spread form tendency tendency Parametric -One-sample t- - 2 - test for a …... -t-test for two -t-test for ….. test population population means independence -test of variance -Paired t-test proportions Parametric -Sign test ….. - 2 - test for -Wilcoxon rank - 2 - test for - 2 -test for Non- goodness of fit sum test or independence homogeneity Mann-Whitney U-Test (Homburg et al. (2008)) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 22 One sample t-Test Objective:  Test assumptions about the population mean µ of a normally distributed variable based on a sample n Guiding question:  Does µ deviate at all from a certain hypothesized mean value µ₀ (two-tailed test)?  Is µ smaller or larger than a certain hypothesized mean value µ₀ (one-tailed test)? Hypothesis:  Two-tailed test: – Null hypothesis H₀: µ = µ₀ – Alternative hypothesis H₁: µ ≠ µ₀ Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 23 One sample t-Test Test statistic: 𝑥:ҧ sample mean 𝑥ҧ − µo µ0: hypothesized mean t= n 𝑠 s: sample standard deviation n: sample size  If H₀ is true, the test statistic follows a Student t-distribution with n – 1 degrees of freedom (df) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 24 4. Calculate the critical value C  Question: At which value of the test statistic do we reject H₀?  Known: Distribution of test statistic if H₀ is true  Wanted: Critical value (C) – Cutoff value that determines when H₀ will be rejected (rejection region) – One-tailed tests: – left: H1: µ < 0; H0: µ ≥ 0  α-percentile of the distribution – right: H1: µ > 0; H0: µ ≤ 0  (1-α)-percentile of the distribution – Two-tailed tests: (1-α/2)-percentile and the (α/2)-percentile – H1: µ ≠ 0; H0: µ = 0  Critical value can be read off the distribution Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 25 4. Calculate the critical value C Left-tailed test Right-tailed test Two-tailed test (Hartmann et al. 2018) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 26 T-distribution table Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 27 5. Calculate the test statistic  Calculate the test statistic for the sample Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 28 6. Draw a conclusion and interpret the results  We then test whether the null hypothesis is false by comparing the test statistic with the critical value(s)  Decision: – Right-tailed (H₁: > 0): rejection of H₀ if test statistics ≥ C – Left-tailed (H₁: < 0): rejection of H₀ if test statistic ≤ -C – Two-tailed tests: Rejection of H₀ if test statistic ≥ C or ≤ -C Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 29  TUM School of Management Technische Universität München Question to you  Example: Customer Satisfaction Survey “Heilbronn South”  A chain of gas stations conducts regular customer satisfaction surveys  Customer satisfaction last year: µ₀ = 2.35 (scale from 1 to 5 stars, 5 stars is best)  Research hypothesis: There should be an increase in customer satisfaction because several initiatives were carried out to improve customer experience  N = 30 customers were surveyed  Mean customer satisfaction this year: 𝑥 = 2.93  Standard deviation: s = 0.91  Test the hypothesis using a t-test statistics! Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 30  TUM School of Management Technische Universität München Question to you  Example: Customer Satisfaction Survey “Heilbronn South”  Step 1:  Null hypothesis (H₀): Customer satisfaction has remained the same or has deteriorated: µCSI ≤ 2.35  Alternative hypothesis (H₁): Customer satisfaction has improved µCSI > 2.35  Step 2:  Significance level α = 0.05  Step 3:  Suitable test procedure: t-test for a population mean  Step 4:  One-tailed test  t-distribution based on the degrees of freedom (df = n-1 = 29)  The 95th percentile of the t-distribution with 29 degrees of freedom is C = 1.699  Decision: Rejection of H₀ if t > 1.699 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 31  TUM School of Management Technische Universität München Question to you  Example: Customer Satisfaction Survey “Heilbronn South”  Step 5:  With:  xത = 2.93  s = 0.91  n=30  µ₀ = 2.35 2.93−2.35  t= 30=3.4910 0.91  Step 6:  t = 3.49 > 1.699: Rejection of H₀  The average customer satisfaction has improved with statistical significance compared to the year before Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 32 Understanding the t-test output of statistical software R: t.text, Stata: ttest 95% Confidence Interval of the Var- Difference P-value Mean iable t df Sig. (2-tailed) Difference Lower Upper CL_T -2.538 29 0.017 -0.567 -1.02 -0.11 Confidence interval for the estimated difference Value of t-statistic Significance level p if null hypothesis is rejected (α) Degrees of freedom of t-statistic (Homburg et al. 2008) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 33 The p-value of the test statistic  p-value: probability of observing an equal or more extreme value of the test statistic than observed, assuming the null hypothesis is true – One-tailed: 𝑝 = Pr 𝑇 ≥ 𝑡 𝐻0 ) or 𝑝 = Pr 𝑇 ≤ 𝑡 𝐻0 ) – Two-tailed: 𝑝 = Pr 𝑎𝑏𝑠(𝑇) ≥ 𝑎𝑏𝑠(𝑡) 𝐻0 )  Interpretation – Null hypothesis is rejected if p < α (test statistic is “statistically significant”) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 34 The p-value of the test statistic  The p-value corresponds to the area under the probability distribution curve  Example t = -2.2611 shaded area = p-value = 0.015713 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 35 Confidence interval  Confidence interval: range of estimates for the parameter  It represents an interval centered on the point estimate (e.g., mean)  Read: “We are (1- α)% confident that the true population parameter is between the lower and upper bound” P(xL ≤ µ ≤ xU) = 1 – α  The lower (xL) and upper (xU) bounds are: sx sx xL = x − z xU = x + z n n  z: percentile of the standard normal distribution corresponding to the significance level α Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 36  Technische Universität München See exercise session “Hypotheses testing”  Technische Universität München Hypothesis testing with regression Regression analysis  Fundamental method of data analysis in almost all scientific disciplines  Key question: How does a change in one or more independent variable(s) affect one dependent variable?  We often want to know how much a change in a variable x changes the variable y x y Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 39 General illustration of a simple linear regression Yi = 0 + 1X1 + i Y Yi Actual 𝜷𝟏 ෢𝒊 Estimated 𝒀 𝜷𝟎 X Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 40 The simple linear regression model  Our notation of the simple linear regression model is: y = 0 + 1x +  Line in a two- dimensional space – 0 is the constant (intercept) – 1 is the coefficient (slope) – y is the dependent variable (or outcome variable) – x is the independent variable (or explanatory variable, or regressors) –  is the error term: all other factors that influence the relationship  Residual: the difference between the actual value of y and the estimated value of y 𝑒𝑖 = 𝑦𝑖 − 𝑦ෝ𝑖 – 𝑦𝑖 = actual value of the dependent variable for observation i – 𝑦ො𝑖 = estimated value of y for observation i as predicted through the regression line Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 41 Regression example Subject 1 2 3 4 5 6 7 8 9 10 Income 34k 87k 80k 67k 75k 65k 34k 91k 62k 76k Happiness 38 89 56 78 72 45 86 81 60 56 Subject 11 12 13 14 15 16 17 18 19 20 Income 39k 74k 68k 59k 89k 56k 54k 56k 72k 57k Happiness 32 54 72 60 87 80 48 37 74 63 Subject 21 22 23 24 25 26 27 28 29 30 Income 86k 40k 89k 69k 56k 79k 69k 73k 55k 69k Happiness 62 46 55 63 51 76 81 66 46 61 Happiness: scale from 0-100 (100 = extremely happy) Income: Euro Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 42 Linear regression example 100 90 80 𝒆𝒊 HAPPINESS (0-100) 70 60 𝜷𝟏 50 40 30 20 𝜷𝟎 HAPPINESSi = 0 + 1INCOMEi + i 10 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 INCOME (EURO) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 43 Linear regression example Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 44 From simple to multiple regression  Bivariate (simple) regression analysis: – 1 independent variable  Multivariate regression analysis: – More than one independent variable – Y = 0 + 1X1 + 2X2 + … + nXn Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 45  TUM School of Management Technische Universität München Question to you  How can we visualize a multivariate regression of the form: 𝑌 = 𝛽0 + 𝛽1 𝑥1 + 𝛽2 𝑥2 𝛽2 𝛽1 𝛽0 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 46 Explained and unexplained variation  To evaluate how well a regression fits the data, we consider the variation it explains Total variation = Explained Variation + Unexplained Variation  Explained variation: difference between the estimated value of y and the mean value of y n Explained variation = ෌i=1(ොyi − yത )2  Unexplained variation: sum of squared residuals n Unexplained variation = ෌i=1(𝑦𝑖 − yො i )2 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 47 Goodness of fit: R2  R2 (R-squared): captures the proportion of variance of the dependent variable explained by the independent variable(s) – Used to describe how “good” the regression explains Y – Also called: coefficient of determination  Idea: Explained Variation Unexplained Variation R2 = =1− Total Variation Total Variation n n ෌i=1(ොyi − yത )2 ෌i=1(yi − yො i )2 = n =1− n ෌i=1(yi − yത )2 ෌i=1(yi − yത )2 – Bounded between 0 and 1 – 0.1 means 10% of variation explained – Rule of thumb: the higher, the more variation explained Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 48  TUM School of Management Technische Universität München Question to you  Which of the following regressions has a higher R²? Estimation (B) Estimation (A) y_2 Fitted values y_1 Fitted values 14 14 12 A 12 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -12 -12 -14 -14 -3.37996 2.87285 -3.37996 2.87285 x x  In both cases 1 = 2  Higher R² in estimation (A) because of less unexplained variance Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 49 Goodness of fit: Adjusted R2  Adjusted R² – Seeks to adjust for mechanical increases in R² when additional regressors are added (1 − R2)(n − 1) Adjusted R2 =1− n−p−1 – n: sample size – p: number of independent variables, excluding the constant – Lower or equal than R² – Negative values possible Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 50 F-test of overall model fit  Statistical test of R2 – H0: All regression coefficients are equal to 0 – H1: At least one regression coefficient is not equal to 0 Explained Variation / p F= Unexplained Variation / (n − p − 1) n ෌i=1(ොyi − yത )2 / p R2 / p F= n = 2 ෌i=1(yi − yො i ) / (n − p − 1) (1 − R2 ) / (n − p − 1) – n: sample size – p: number of independent variables, excluding the constant  Idea: tests whether the estimated model fits the data better than a model with no independent variables – Adjusts for the number of variables in the model Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 51 t-Test for single regression coefficients  Statistical test of regression coefficients – H0: β = 0 – H1: β ≠ 0  One-tailed formulation is possible  Formal specification β 𝑡= s𝑖 – s𝑖 : standard error (i.e., estimated standard deviation) of regression coefficient  If H0 is true, the test statistic follows a t-distribution with (n-p-1) degrees of freedom (df) Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 52 Interpreting regression coefficients Functional form Effect size interpretation (where ß is the coefficient Linear f A unit change in x is associated with an average y = f(x) change of ß units in y. ln(y) = f(x) For a unit increase in x, y increases on average by the percentage 100 𝑒 ß − 1 (≅ 100 ß 𝑤ℎ𝑒𝑛 ß < y = f(ln(x)) For 1% increase in x, y increases on average by ln 1.01 × ß (≅ ß/100). ln(y) = f(ln(x)) For a 1% increase in x, y increases on average by the percentage 100 𝑒 ß ×ln 1.01 − 1 (≅ ß 𝑤ℎ𝑒𝑛 ß < 0.1). Lin et al. (2013), p. 906-917 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 53 Reporting regression results Lin et al. (2013), p. 906-917 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 54 Estimating the regression model: the OLS approach  Different estimators – Ordinary least squares (OLS) – Partial least squares (PLS) – Generalized least squares (GLS) – …  Ordinary least squares (OLS) – Minimization of the squared distance between the regression line and y values (residuals) – Estimates the coefficients  in the regression model by: 𝑛 min(𝑈𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛) = 𝑚𝑖𝑛 ෍ 𝑒𝑖2 𝑖=1 – 𝑒𝑖2 = squared residual for observation i – Squared to account for positive and negative residuals Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 55  Technische Universität München See exercise session “Regressions” Moderation Happiness Money Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 57 Idea of moderation Moderation model Possible Outcomes y High level of x2 X2 Low level of x2 X1 Y x1 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 58 Moderation test  Moderation is modeled via product terms Y = 0 + 1X1 + 2X2 + 3(X1 X2)+  X2 – X2 is called the moderator – X1 X2 is called the interaction term X1 Y – 1 and 2 are called main effects – 3 is called the moderating (interaction) effect  Examples of formulations for moderation hypotheses – Relationship between X1 and Y differs depending on X2 – X2 modifies the effect of X1 and Y – X1 and Y’s relationship is different at different levels of X2 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 59 Types of moderation effects Happiness  X1 continuous, X2 categorical – Slope of the relationship between Y and X1 differs across the groups represented by X2 Money  X1 continuous, X2 continuous Happiness – Slope of the relationship between Y and X1 varies (i.e., increases or decreases) according to the level of X2 – Discretize one of the categorical variables Money  X1 categorical, X2 categorical – Difference between the group means for X1 differ depending on group membership X2 Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 60 Types of moderation effects  Special case: U-shape / inverted U-shape – Y = 0 + 1X1 + 2(X1 X1)+  Happiness Money Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 61 Questions Professorship for Innovation & Digitalization (Prof. Dr. Jens Förderer) 62