Multilevel Modeling -7

Multilevel Moderation Moderation in MLM • Three types of moderation in 2 level model: • Level 1 x Level 2 = cross-level interaction • Level 1 x Level 1 within-level interaction • Level 2 x Level 2 • Cross-level interactions seem to be the most common, there are fewer resources about how to conduct and probe within-level interactions Within-Level Moderation in MLM • Level 2 interactions: • Similar to moderation in linear regression • Level 1 interactions: • Must decide whether to person-mean center or grand-mean center each predictor • interpret results accordingly • Must decide whether to allow any random effects • If no random effects, can treat like traditional regression in terms of probing simple slopes, etc • If allow any random effects, more complicated to probe interaction (See link to calculator on next slide) • Recommendation by Barr 2013 to allow a random slope for interaction term • Recommendation – Test for significance of random effects. Ask: Does it make sense theoretically to allow individual variation? Does it change the estimates when allowing random effects? Level 1 Moderation Issues • If you have a moderation between two level 1 predictors and there are random slopes, then unlike OLS linear regression your interaction estimate may change depending on how your predictors are centered (i.e., the value will be different if you change the meaning of 0). Thus, have to be thoughtful in deciding how to center data and mindful if probing simple slopes • For probing simple slopes: can re-center fixed effects but keep original centering for random effect. This will prevent the interaction term from changing. • Level 1 Moderation Issues Here is an interaction between conflict and perspective-taking. Conflict is z-scored in the analysis on the left but not on the right, and we get different effects for interaction term and random slopes when conflict is z-scored versus not z-scored -this is due to what 0 represents, not change in SD, same exact results if group-mean center Note: Effects will look exactly like those for the z-scored conflict if you enter raw conflict as fixed effect and z-scored conflict as random effect. Moderation Online calculator for probing all three types of interactions: http://www.quantpsy.org/interact/hlm2.htm Info needed: -regression coefficients -dfs for intercept and slope -variance covariance matrix (typically not default output but can be requested) Note: calculators assume level 1 predictors allowed to be random, can use more traditional methods when level 1 predictors are fixed Moderation • Want to do it by hand? Use Aiken and West’s approach to probing interactions by re-centering your main effects and re-computing your interaction term accordingly • If have random slopes in model, need to keep original variable as random effect and only re-center fixed effect. Example: • Self-Focus x Sleep interaction predicting gratitude. Does self-focus enhance the negative effects of sleep on gratitude? • Effect of gratitude at +1 SD of self-focus • Re-center self-focus by SUBTRACTING 1 SD so that 0 now represents +1 SD • Create new interaction term with mean-centered gratitude and +1 SD centered partner support • Rerun model with new terms • Interpret main effect of gratitude as gratitude for people +1 SD above mean on partner support • Repeat for -1 SD by ADDING 1 SD to partner support Resources for Moderation & Mediation • General resources (both of these sites are great sources of info!): • http://davidakenny.net/cm/mediate.htm • http://www.quantpsy.org/medn.htm • Moderation in MLM • Bauer & Curran article on probing multilevel moderations: http://www.unc.edu/~curran/pdfs/Bauer%26Curran(2005).pdf • Barr article on random effects: click here • Calculators: http://www.quantpsy.org/interact/hlm2.htm • Mediation in MLM • Readable slides from Chris Preacher on multilevel mediation: http://afhayes.com/public/aps2013.pdf (this has many additional references at the end) • Zhang, Z., Zypher, M. J., & Preacher, K. J. (2009). Testing multilevel mediation using hierarchical linear models: Problems and solutions. Organizational Research Methods, 12, 695-719. From prior slides: • http://quantpsy.org/pubs/bauer_preacher_gil_2006.pdf (1 model approach to MLM mediation) • https://njrockwood.com/mlmed/ (multilevel mediation macro for SPSS) • http://www.quantpsy.org/pubs/preacher_zhang_zyphur_2011.pdf (multilevel mediation in SEM) Calculators from prior slides: Calculator if 1 or 0 random effects: http://www.quantpsy.org/medmc/medmc.htm Calculator for 1-1-1 with random effects: http://www.quantpsy.org/medmc/medmc111.htm Time-Lagged Analyses • With data measured over a span of time, such as diary data, you might want to know if a predictor on day t influences an outcome on day t+1. • Could also be thought of as day t-1 and day t • Could do this with day t and day t+2, t+3, and so on • Does stress one day reduce SWL the next day? • Create Lagged Day Variables (some programs have a function for this) • This will create a new set of variables that are day t-1 • Regress outcome on day t+1 onto predictor and outcome on day t. • Tomorrow’s SWL is predicted by today’s stress and today’s SWL • Could also refer to this as “Today’s SWL predicted by yesterday’s stress and SWL,” it’s the same thing • Controlling for outcome today is important for making sure you aren’t just tapping into effect of predictor on outcome that day Time-Lagged Analyses • Can also do a sort of cheater lagged analysis in which you predict today’s outcome from today’s predictor controlling for yesterday’s outcome • OutcomeT = PredictorT + OutcomeT-1 • This type of analysis predicts changes in outcome from yesterday while looking at relationship between predictor and outcome today Time-Lagged Analyses • Word of warning: • If you try to center variables at Level 1 and then do lagged-day analyses, these can get really funky if you have a small number of Level 1 observations (e.g., 7-day diary) because your variables are fluctuations around the mean so if a variable is high one day, it by definition has to be low another day, often leading to negative relationships between variables from one day to the next. T1 T2 T2 T1 T1 T2 T2 T2 T1 T1 T1 T2 T1 T2 T2 T2 T2 T2 T1 T1 T1 T1 Growth-Curve Analyses • Can look at how variables change over time in diary data, longitudinal designs. • Create variable to capture time • E.g., DAYS = 1-14 of 14 day diary • Make intercept meaningful by recoding so Day 1 = 0 • DAYS = 0-13 • Can look at curve of slope by creating DAYS2, DAYS3 • Follow basic principles for looking at higher order associations Growth-Curve Analyses • Can look at how variables change over time in diary data, longitudinal designs. • Use time variable (e.g., DAYS) as a predictor • Does gratitude increase across two week diary? • Gratitude as outcome, DAYS as predictor • Intercept = level of gratitude at start of diary • DAYS slope = change in gratitude across two weeks • Does gratitude increase more for people in gratitude intervention? • Look at interaction between DAYS and CONDITION predicting gratitude • Is slope between days and gratitude different for people in intervention versus control conditions? Growth-Curve Analyses • Great resources for GCA in R: • http://www.danmirman.org/gca • Book: Mirman, D. (2014). Growth Curve Analysis and Visualization Using R. Chapman and Hall / CRC. Power & Sample Size Calculations in MLM It can be complicated. There is more out there than what I am presenting. But this should help you be aware of the issues you’ll face. Calculating Power/Sample Size in MLM • Have to consider power/sample size for each level of data • How many participants/groups to collect? • How many days/members of group to collect? (i.e., cluster size) • Gain more power by increasing upper level units • 100 participants with 5 days of data = more power • 50 participants with 10 days of data • ICC matters: if ICC is very low, then increasing the cluster size is useful, if very high then additional days/members of group etc. are not useful Calculating Power/Sample Size in MLM • One approach is using the DESIGN EFFECT: neffective = n/[1+(ncluster - 1)*] • neffective = effective sample size • n= total sample size • ncluster = number of days/people in group etc. (e.g.,7 for a week-long diary) •  = ICC for that variable This tells you for a given sample size, what your effective sample size is. Ex: If  = 1 (no new info gained from additional level 1 units), then sample size is number of level 2 units. Ex: If  = 0 (no actual nesting/non-independence between level 1 units), then sample size is number of level 1 units. In actuality, will likely end up somewhere in between • To figure out needed sample size, use ICC from pilot testing/prior research to determine total sample size needed given effective sample size n= neffective(1+(ncluster - 1)*] Calculating Power/Sample Size in MLM • PINT: calculator for power in two level models • https://www.stats.ox.ac.uk/~snijders/multilevel.htm Written by Tom Snijders, Roel Bosker, and Henk Guldemond Bootstrap results from pilot data (seems like there are some movements to develop this approach). Power is the percent of times the results are significant for the given parameters (effect size, sample size, ICC, etc). Bolger, N., Stadler, G., & Laurenceau, J.-P. (2012). Power analysis for intensive longitudinal studies. In M. R. Mehl & T. S. Conner (Eds.), Handbook of research methods for studying daily life (pp. 285-301). New York, NY, US: The Guilford Press. A Few Online Resources for Multilevel Modeling • UCLA Resources intro to MLM: • https://stats.idre.ucla.edu/other/mult-pkg/introduction-to-linear-mixed-models/ • The Analysis Factor • https://www.theanalysisfactor.com/resources/by-topic/mixed-multilevel-models/ • Dave Kenny’s Website (great for dyadic data): • http://davidakenny.net/ • Conducting MLM analyses in R: • http://cran.r-project.org/doc/contrib/Bliese_Multilevel.pdf • https://www.jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r • Guide to Linear Mixed Models in SPSS: • http://www.spss.ch/upload/1126184451_Linear%20Mixed%20Effects%20Modeling%20in%20SPSS.pdf • A recent Psych Methods article with practical advice for novices: • http://psycnet.apa.org/doiLanding?doi=10.1037%2Fmet0000159 • Recommended Books: • Multilevel Analysis: Techniques and Applications by Joop Hox • Intensive Longitudinal Methods (Bolger & Laurenceau, 2013)