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Mixed Effects Models - Growth Curve Analysis

S
Scott Fraundorf

The document outlines preparations for a course on growth curve analysis, detailing final project expectations, including data analysis and in-class presentations. It covers methods for analyzing longitudinal data, particularly through mixed-effects models, emphasizing the importance of a hierarchical data structure and various modeling techniques. Additionally, it discusses the interaction of time as a predictor variable and how language acquisition influences vocabulary growth in children, with practical examples of reshaping data formats in R.

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Course Business ! Lab materials on Canvas ! Final project information will be posted on Canvas at 4:00 PM today • Final project: Analyze your own data • I can supply simulated data if needed—send me an intro & methods section for the data you’d like to analyze • In-class presentation: Last 2.5 weeks of class, beginning Nov. 9th • Sign up on Canvas for a time slot – first come, first served! • Final paper: Due on Canvas last day of class • Canvas will have more info. on requirements
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
Longitudinal Designs Time
Longitudinal Designs • There are many methods available for analyzing longitudinal data • Aidan Wright’s class on Applied Longitudinal Data Analysis • Mixed-effects models are one effective solution • Hierarchical models naturally account for longitudinal data • Include all of the other features we’ve looked at (multiple random effects, mix of categorical & continuous variables, non-normal DVs, etc.)
School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 • What kind of random-effects structure is this? Longitudinal Data as Hierarchical Data
School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 • What kind of random-effects structure is this? • Two levels of nesting – sample schools, then sample students inside each school Longitudinal Data as Hierarchical Data
Student 1 Exam 1 Student 2 Exam 2 Student 3 Exam 1 Student 3 Exam 2 • Now imagine we observed each student several different times • e.g., midterm & final exam School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 Longitudinal Data as Hierarchical Data
Student 1 Exam 1 Student 2 Exam 2 Student 3 Exam 1 Student 3 Exam 2 School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 3 LEVEL 2 • This is just another level of nesting • Sample schools • Sample student within each school • Sample time points within each student Longitudinal Data as Hierarchical Data Sampled TIME POINTS LEVEL 1
Longitudinal Designs • Two big questions mixed-effects models can help us answer about longitudinal data: 1) What is the overall trajectory of change across time? " Today 2) How does an observation at one time point relate to the next time point? " Wednesday
vocab.csv ! Role of language input in early vocab acquisition ! 200 kids from lower SES families: ! 100 families given books to read to their kids ! 100 waitlisted controls ! Vocab assessed every 2 months from 20 mos. to 28 mos. ! Research questions: ! What is the general trajectory of vocabulary acquisition in these ages? ! How is this altered by our intervention?
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
Long vs. Wide Format ! For lmer(), each observation needs its own row ! “long” format Time 1 row Time 2 gets a separate row One trial Another trial
Long vs. Wide Format ! For lmer(), each observation needs its own row ! “long” format ! Sometimes data comes to us in “wide” format ! Each repeated measure is a different column in the same row ! e.g., what SPSS uses ! vocab.csv is in this format Time 1 and Time 2 are considered separate variables Time 1 row Time 2 gets a separate row
Long vs. Wide Format ! Tidyverse contains very easy functions for pivoting data between formats ! pivot_longer(): To turn data into long format ! pivot_wider(): To turn data into wide format
pivot_longer() ! Here is a look at our dataframe… We want to turn into each of these columns into a separate row
pivot_longer() ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8')) -> vocab.p ! Now we have five rows per child We want to turn into each of these columns into a separate row
pivot_longer() ! By default, R saves the original column name in a variable called “name”, and the DV in a variable called “value” ! Not very informative ! We can specify better names : ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8’), names_to='MonthsInStudy’, values_to='VocabWords') -> vocab.p
pivot_longer() ! By default, R saves the original column name in a variable called “name”, and the DV in a variable called “value” ! Not very informative ! We can specify better names : ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8’), names_to='MonthsInStudy’, values_to='VocabWords') -> vocab.p
pivot_longer() ! What we’d really like is to turn MonthsIntoStudy into a numeric variable ! Currently, the values are “Month0,” “Month2,” etc., because those were the original column names ! Since these all start with the same prefix (Month), tidyverse can automatically strip that out for us ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4',' Month6','Month8’), names_to='MonthsInStudy', names_prefix='Month', values_to='VocabWords')-> vocab.p
pivot_longer() ! What we’d really like is to turn MonthsIntoStudy into a numeric variable ! Currently, the values are “Month0,” “Month2,” etc., because those were the original column names ! Since these all start with the same prefix (Month), tidyverse can automatically strip that out for us ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4',' Month6','Month8’), names_to='MonthsInStudy', names_prefix='Month', values_to='VocabWords')-> vocab.p
pivot_longer() ! Now, we can use as.numeric() on this variable ! vocab.p %>% mutate(MonthsInStudy=as.numeric(MonthsInStudy)) -> vocab.p
pivot_longer() ! Now, we can use as.numeric() on this variable ! vocab.p %>% mutate(MonthsInStudy=as.numeric(MonthsInStudy)) -> vocab.p
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
vocab.csv ! Role of language input in early vocab acquisition ! 200 kids from lower SES families: ! 100 families given books to read to their kids ! 100 waitlisted controls ! Vocab assessed every 2 months from 20 mos. to 28 mos. ! Research questions: ! What is the general trajectory of vocabulary acquisition in these ages? ! How is this altered by our intervention?
Longitudinal Designs • Two big questions mixed-effects models can help us answer about longitudinal data: 1) What is the overall trajectory of change across time? " Today: Growth curve analysis 2) How does an observation at one time point relate to the next time point? " Wednesday
Time as a Predictor Variable • How does vocabulary change over time? • The simple way to answer this question: Add time as a variable in our model • Nothing “special” about time as a predictor
Time as a Predictor Variable • Let’s add the effect of time to our model • Here: MonthsInStudy (starting at 0) • Fixed effect because we’re interested in time effects • model1 <- lmer(VocabWords ~ 1 + MonthsInStudy + (1|Child), data=vocab) • How would you interpret the two estimates? • Intercept: • Slope:
Time as a Predictor Variable • Let’s add the effect of time to our model • Here: MonthsInStudy (starting at 0) • Fixed effect because we’re interested in time effects • model1 <- lmer(VocabWords ~ 1 + MonthsInStudy + (1|Child), data=vocab) • How would you interpret the two estimates? • Intercept: Average vocab ~51 words at start of study • Slope: Gain of about ~35 words per month
Time as a Predictor Variable • Not necessary to have every time point represented • Don’t even have to be the same set of time points across participants • Time units also don’t matter as long as they’re consistent • Could be hours, days, years …
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
Quadratic & Higher Degrees
Quadratic & Higher Degrees • We’ve been assuming a linear effect of time
Quadratic & Higher Degrees • But, it looks like vocab growth may accelerate • Growth between 0 mo. and 2 mo. is much smaller than growth between 6 mo. and 8 mo. • Suggests a curve / quadratic equation
Quadratic & Higher Degrees • Add quadratic effect (MonthsInStudy2): • model.poly <- lmer(VocabWords ~ 1 + poly(MonthsInStudy,degree=2,raw=TRUE) + (1|Child), data=vocab.p) • degree=2 because we want MonthsInStudy2 • poly() automatically adds lower-order terms as well • i.e., the linear term (MonthsInStudy1) • raw=TRUE to keep the original scale of the variables (time measured in months)
Quadratic & Higher Degrees • Results: • Implied equation (approximate): • VocabWords = 75 + 11*Months + 3*Months2 • What are predicted values if… • Month=0? • Month=1? • Month=2? Intercept Linear term Quadratic term
Quadratic & Higher Degrees • Results: • Implied equation (approximate): • VocabWords = 75 + 11*Months + 3*Months2 • What are predicted values if… • Month=0? • Month=1? • Month=2? • Vocab growth is accelerating (larger change from month 1 to month 2 than from month 0 to month 1) Intercept Linear term Quadratic term VocabWords=75+(11*0)+(3*02) = 75 VocabWords=75+(11*1)+(3*12) = 89 VocabWords=75+(11*2)+(3*22) = 109 +20 +14
POSITIVE QUADRATIC TREND & NO LINEAR TREND NEGATIVE QUADRATIC TREND & NO LINEAR TREND POSITIVE QUADRATIC TREND & POSITIVE LINEAR TREND NEGATIVE QUADRATIC TREND & POSITIVE LINEAR TREND Different patterns of quadratic & linear effects in a GCA describe different curves
POSITIVE QUADRATIC TREND & NO LINEAR TREND NEGATIVE QUADRATIC TREND & NO LINEAR TREND POSITIVE QUADRATIC TREND & POSITIVE LINEAR TREND NEGATIVE QUADRATIC TREND & POSITIVE LINEAR TREND • Linear trend: Is there an overall increase over time (+), decrease (-), or no change? • Quadratic trend: Is the line curved up (+), down (-), or straight?
Quadratic & Higher Degrees • The Yerkes-Dodson law (Yerkes & Dodson, 1908) describes the optimal level of physiological arousal to perform a task: Low arousal results in poor performance because you are not alert enough, medium arousal results in strong performance, and high arousal results in poor performance because you are too anxious. Linear trend: Quadratic trend:
Quadratic & Higher Degrees • The Yerkes-Dodson law (Yerkes & Dodson, 1908) describes the optimal level of physiological arousal to perform a task: Low arousal results in poor performance because you are not alert enough, medium arousal results in strong performance, and high arousal results in poor performance because you are too anxious. Linear trend: Quadratic trend: NONE NEGATIVE
Quadratic & Higher Degrees • In adulthood, working memory declines the older you get (Park et al., 2002). Linear trend: Quadratic trend:
Quadratic & Higher Degrees • In adulthood, working memory declines the older you get (Park et al., 2002). Linear trend: Quadratic trend: NONE NEGATIVE
Quadratic & Higher Degrees • Aphasia is a neuropsychological disorder that disrupts speech, often resulting from a stroke. After initially acquiring aphasia, patients often experience rapid recovery in much of their language performance (dependent variable) as time (independent variable) increases. But, this recovery eventually slows down, and language performance doesn’t get much better no matter how much more time increases (e.g., Demeurisse et al., 1980). Linear trend: Quadratic trend:
Quadratic & Higher Degrees • Aphasia is a neuropsychological disorder that disrupts speech, often resulting from a stroke. After initially acquiring aphasia, patients often experience rapid recovery in much of their language performance (dependent variable) as time (independent variable) increases. But, this recovery eventually slows down, and language performance doesn’t get much better no matter how much more time increases (e.g., Demeurisse et al., 1980). Linear trend: Quadratic trend: POSITIVE NEGATIVE
Quadratic & Higher Degrees • Studies of practice and expertise (e.g., Logan, 1988) show that people learning to do a task—such as arithmetic—initially show a quick decrease in response time (dependent variable) as the amount of practice increases. However, eventually they hit the point where the task can’t possibly be done any faster, and response time reaches an asymptote and stops decreasing. Linear trend: Quadratic trend:
Quadratic & Higher Degrees • Studies of practice and expertise (e.g., Logan, 1988) show that people learning to do a task—such as arithmetic—initially show a quick decrease in response time (dependent variable) as the amount of practice increases. However, eventually they hit the point where the task can’t possibly be done any faster, and response time reaches an asymptote and stops decreasing. Linear trend: Quadratic trend: NEGATIVE POSITIVE
Quadratic & Higher Degrees • Could go up to even higher degrees (Time3, Time4…) • degree=3 if highest exponent is 3 • Degree minus 1 = Number of bends in the curve -100 -50 0 50 100 x^3 0 20 40 60 80 100 x^1 0 20 40 60 80 100 x^2 0 bends 1 bend 2 bends
Quadratic & Higher Degrees • Maximum degree of polynomial: # of time points minus 1 • Example: 2 time points perfectly fit by a line (degree 1). Nothing left for a quadratic term to explain. • But, don’t want to overfit • Probably not the case that the real underlying (population) trajectory has 6 bends in it • What degree should we include? • Theoretical considerations • If comparing conditions, look at mean trajectory across conditions (Mirman et al., 2008)
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
Child 2 Child 1 Child 3 Child 4 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Child 3 Assess- ment 1 Child 3 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 • So far, we assume the same growth rate for all kids • Almost certainly not true! • At level 2, we’re sampling kids both with different starting points (intercepts) and growth rates (slopes) Longitudinal Data: Random Slopes Growth Rate 1 Growth Rate 2
Longitudinal Data: Random Slopes RANDOM INTERCEPTS MODEL Kids vary in starting point, but all acquire vocabulary at the same rate over this period WITH RANDOM SLOPES Allows rate of vocab acquisition to vary across kids (as well as intercept)
• Let’s allow the MonthsInStudy effect to be different for each Child • model.Slope <- lmer(VocabWords ~ 1 + poly(MonthsInStudy,degree=2,raw=TRUE) + (1 + poly(MonthsInStudy,degree=2,raw=TRUE)|Child), data=vocab.p) Longitudinal Data: Random Slopes Child 2 Child 1 Child 3 Child 4 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Child 3 Assess- ment 1 Child 3 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 Growth Rate 1 Growth Rate 2
Longitudinal Data: Random Slopes Substantial variability in the MonthsInStudy slope SD of the slope across children is ~5 words Mean slope is 11 words/mo, but some kids might have a slope of 6 or 16
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
Time-Variant & -Invariant Variables • We may want to include other (experimental or observational) variables in a GCA model: • e.g., our reading intervention • MonthsInStudy + Reading • Effect of Reading invariant across time • Can only affect the intercept (parallel lines) • MonthsInStudy * Reading • Effect of Reading varies with time • Can affect intercept & slope
Time-Variant & -Invariant Variables • Fixed-effect results with the interaction: e.g., Huttenlocher et al., 1991 Also results in faster vocab growth (amplifies + Time effect) Linear growth rate for “No” group: 8.4 words / month Linear growth rate for “Yes” group: 8.4 + 4.7 = 13.1 words / month Effect of reading at time 0 (intercept): ~6 words No effect on curvature (quadratic term)
Time-Variant & -Invariant Variables • Can be either: • Time-Invariant Predictor: Same across all time points within a subject • e.g., our intervention, race/ethnicity • Level 2 variables • Time-Varying Predictor: Varies even within a subject, from one time point to another • e.g., hours of sleep, mood • Level-1 variable Child 2 Child 1 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 • Since R automatically figures out what’s a level-1 vs. level-2 variable, we don’t have to do anything special for either kind of variable
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
! Another possibility (in general) is a qualitative change in the trajectory at some point in the range of time observed ! e.g., age of majority, public policy changes, etc. ! Include a breakpoint—a categorical variable that represents whether or not you’re above the point at which the change happens ! mydata %>% mutate(Year2015OrLater= ifelse(Year >= 2015, 1, 0))-> mydata ! New variable is: ! 0 before 2015 ! 1 for 2015 onwards Breakpoints
! Another possibility (in general) is a qualitative change in the trajectory at some point in the range of time observed ! e.g., age of majority, public policy changes, etc. ! Include a breakpoint—a categorical variable that represents whether or not you’re above the point at which the change happens Breakpoints Main effect of breakpoint only – single shift downward (or upward) but same slope Main effect of breakpoint & an interaction – slope also changes
! Can apply this technique to any continuous variables, not just time ! e.g., above/below the poverty line ! As with higher-order polynomials, be wary of overfitting the data ! Most effective if we have an a priori reason to expect a breakpoint somewhere ! Or, use cross-validation to support a more exploratory analysis Breakpoints
Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab

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Mixed Effects Models - Growth Curve Analysis

  • 1. Course Business ! Lab materials on Canvas ! Final project information will be posted on Canvas at 4:00 PM today • Final project: Analyze your own data • I can supply simulated data if needed—send me an intro & methods section for the data you’d like to analyze • In-class presentation: Last 2.5 weeks of class, beginning Nov. 9th • Sign up on Canvas for a time slot – first come, first served! • Final paper: Due on Canvas last day of class • Canvas will have more info. on requirements
  • 2. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 4. Longitudinal Designs • There are many methods available for analyzing longitudinal data • Aidan Wright’s class on Applied Longitudinal Data Analysis • Mixed-effects models are one effective solution • Hierarchical models naturally account for longitudinal data • Include all of the other features we’ve looked at (multiple random effects, mix of categorical & continuous variables, non-normal DVs, etc.)
  • 5. School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 • What kind of random-effects structure is this? Longitudinal Data as Hierarchical Data
  • 6. School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 • What kind of random-effects structure is this? • Two levels of nesting – sample schools, then sample students inside each school Longitudinal Data as Hierarchical Data
  • 7. Student 1 Exam 1 Student 2 Exam 2 Student 3 Exam 1 Student 3 Exam 2 • Now imagine we observed each student several different times • e.g., midterm & final exam School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 2 LEVEL 1 Longitudinal Data as Hierarchical Data
  • 8. Student 1 Exam 1 Student 2 Exam 2 Student 3 Exam 1 Student 3 Exam 2 School 1 School 2 Student 2 Student 1 Student 3 Student 4 Sampled SCHOOLS Sampled STUDENTS LEVEL 3 LEVEL 2 • This is just another level of nesting • Sample schools • Sample student within each school • Sample time points within each student Longitudinal Data as Hierarchical Data Sampled TIME POINTS LEVEL 1
  • 9. Longitudinal Designs • Two big questions mixed-effects models can help us answer about longitudinal data: 1) What is the overall trajectory of change across time? " Today 2) How does an observation at one time point relate to the next time point? " Wednesday
  • 10. vocab.csv ! Role of language input in early vocab acquisition ! 200 kids from lower SES families: ! 100 families given books to read to their kids ! 100 waitlisted controls ! Vocab assessed every 2 months from 20 mos. to 28 mos. ! Research questions: ! What is the general trajectory of vocabulary acquisition in these ages? ! How is this altered by our intervention?
  • 11. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 12. Long vs. Wide Format ! For lmer(), each observation needs its own row ! “long” format Time 1 row Time 2 gets a separate row One trial Another trial
  • 13. Long vs. Wide Format ! For lmer(), each observation needs its own row ! “long” format ! Sometimes data comes to us in “wide” format ! Each repeated measure is a different column in the same row ! e.g., what SPSS uses ! vocab.csv is in this format Time 1 and Time 2 are considered separate variables Time 1 row Time 2 gets a separate row
  • 14. Long vs. Wide Format ! Tidyverse contains very easy functions for pivoting data between formats ! pivot_longer(): To turn data into long format ! pivot_wider(): To turn data into wide format
  • 15. pivot_longer() ! Here is a look at our dataframe… We want to turn into each of these columns into a separate row
  • 16. pivot_longer() ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8')) -> vocab.p ! Now we have five rows per child We want to turn into each of these columns into a separate row
  • 17. pivot_longer() ! By default, R saves the original column name in a variable called “name”, and the DV in a variable called “value” ! Not very informative ! We can specify better names : ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8’), names_to='MonthsInStudy’, values_to='VocabWords') -> vocab.p
  • 18. pivot_longer() ! By default, R saves the original column name in a variable called “name”, and the DV in a variable called “value” ! Not very informative ! We can specify better names : ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4’, 'Month6','Month8’), names_to='MonthsInStudy’, values_to='VocabWords') -> vocab.p
  • 19. pivot_longer() ! What we’d really like is to turn MonthsIntoStudy into a numeric variable ! Currently, the values are “Month0,” “Month2,” etc., because those were the original column names ! Since these all start with the same prefix (Month), tidyverse can automatically strip that out for us ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4',' Month6','Month8’), names_to='MonthsInStudy', names_prefix='Month', values_to='VocabWords')-> vocab.p
  • 20. pivot_longer() ! What we’d really like is to turn MonthsIntoStudy into a numeric variable ! Currently, the values are “Month0,” “Month2,” etc., because those were the original column names ! Since these all start with the same prefix (Month), tidyverse can automatically strip that out for us ! vocab %>% pivot_longer(cols=c('Month0','Month2','Month4',' Month6','Month8’), names_to='MonthsInStudy', names_prefix='Month', values_to='VocabWords')-> vocab.p
  • 21. pivot_longer() ! Now, we can use as.numeric() on this variable ! vocab.p %>% mutate(MonthsInStudy=as.numeric(MonthsInStudy)) -> vocab.p
  • 22. pivot_longer() ! Now, we can use as.numeric() on this variable ! vocab.p %>% mutate(MonthsInStudy=as.numeric(MonthsInStudy)) -> vocab.p
  • 23. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 24. vocab.csv ! Role of language input in early vocab acquisition ! 200 kids from lower SES families: ! 100 families given books to read to their kids ! 100 waitlisted controls ! Vocab assessed every 2 months from 20 mos. to 28 mos. ! Research questions: ! What is the general trajectory of vocabulary acquisition in these ages? ! How is this altered by our intervention?
  • 25. Longitudinal Designs • Two big questions mixed-effects models can help us answer about longitudinal data: 1) What is the overall trajectory of change across time? " Today: Growth curve analysis 2) How does an observation at one time point relate to the next time point? " Wednesday
  • 26. Time as a Predictor Variable • How does vocabulary change over time? • The simple way to answer this question: Add time as a variable in our model • Nothing “special” about time as a predictor
  • 27. Time as a Predictor Variable • Let’s add the effect of time to our model • Here: MonthsInStudy (starting at 0) • Fixed effect because we’re interested in time effects • model1 <- lmer(VocabWords ~ 1 + MonthsInStudy + (1|Child), data=vocab) • How would you interpret the two estimates? • Intercept: • Slope:
  • 28. Time as a Predictor Variable • Let’s add the effect of time to our model • Here: MonthsInStudy (starting at 0) • Fixed effect because we’re interested in time effects • model1 <- lmer(VocabWords ~ 1 + MonthsInStudy + (1|Child), data=vocab) • How would you interpret the two estimates? • Intercept: Average vocab ~51 words at start of study • Slope: Gain of about ~35 words per month
  • 29. Time as a Predictor Variable • Not necessary to have every time point represented • Don’t even have to be the same set of time points across participants • Time units also don’t matter as long as they’re consistent • Could be hours, days, years …
  • 30. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 32. Quadratic & Higher Degrees • We’ve been assuming a linear effect of time
  • 33. Quadratic & Higher Degrees • But, it looks like vocab growth may accelerate • Growth between 0 mo. and 2 mo. is much smaller than growth between 6 mo. and 8 mo. • Suggests a curve / quadratic equation
  • 34. Quadratic & Higher Degrees • Add quadratic effect (MonthsInStudy2): • model.poly <- lmer(VocabWords ~ 1 + poly(MonthsInStudy,degree=2,raw=TRUE) + (1|Child), data=vocab.p) • degree=2 because we want MonthsInStudy2 • poly() automatically adds lower-order terms as well • i.e., the linear term (MonthsInStudy1) • raw=TRUE to keep the original scale of the variables (time measured in months)
  • 35. Quadratic & Higher Degrees • Results: • Implied equation (approximate): • VocabWords = 75 + 11*Months + 3*Months2 • What are predicted values if… • Month=0? • Month=1? • Month=2? Intercept Linear term Quadratic term
  • 36. Quadratic & Higher Degrees • Results: • Implied equation (approximate): • VocabWords = 75 + 11*Months + 3*Months2 • What are predicted values if… • Month=0? • Month=1? • Month=2? • Vocab growth is accelerating (larger change from month 1 to month 2 than from month 0 to month 1) Intercept Linear term Quadratic term VocabWords=75+(11*0)+(3*02) = 75 VocabWords=75+(11*1)+(3*12) = 89 VocabWords=75+(11*2)+(3*22) = 109 +20 +14
  • 37. POSITIVE QUADRATIC TREND & NO LINEAR TREND NEGATIVE QUADRATIC TREND & NO LINEAR TREND POSITIVE QUADRATIC TREND & POSITIVE LINEAR TREND NEGATIVE QUADRATIC TREND & POSITIVE LINEAR TREND Different patterns of quadratic & linear effects in a GCA describe different curves
  • 38. POSITIVE QUADRATIC TREND & NO LINEAR TREND NEGATIVE QUADRATIC TREND & NO LINEAR TREND POSITIVE QUADRATIC TREND & POSITIVE LINEAR TREND NEGATIVE QUADRATIC TREND & POSITIVE LINEAR TREND • Linear trend: Is there an overall increase over time (+), decrease (-), or no change? • Quadratic trend: Is the line curved up (+), down (-), or straight?
  • 39. Quadratic & Higher Degrees • The Yerkes-Dodson law (Yerkes & Dodson, 1908) describes the optimal level of physiological arousal to perform a task: Low arousal results in poor performance because you are not alert enough, medium arousal results in strong performance, and high arousal results in poor performance because you are too anxious. Linear trend: Quadratic trend:
  • 40. Quadratic & Higher Degrees • The Yerkes-Dodson law (Yerkes & Dodson, 1908) describes the optimal level of physiological arousal to perform a task: Low arousal results in poor performance because you are not alert enough, medium arousal results in strong performance, and high arousal results in poor performance because you are too anxious. Linear trend: Quadratic trend: NONE NEGATIVE
  • 41. Quadratic & Higher Degrees • In adulthood, working memory declines the older you get (Park et al., 2002). Linear trend: Quadratic trend:
  • 42. Quadratic & Higher Degrees • In adulthood, working memory declines the older you get (Park et al., 2002). Linear trend: Quadratic trend: NONE NEGATIVE
  • 43. Quadratic & Higher Degrees • Aphasia is a neuropsychological disorder that disrupts speech, often resulting from a stroke. After initially acquiring aphasia, patients often experience rapid recovery in much of their language performance (dependent variable) as time (independent variable) increases. But, this recovery eventually slows down, and language performance doesn’t get much better no matter how much more time increases (e.g., Demeurisse et al., 1980). Linear trend: Quadratic trend:
  • 44. Quadratic & Higher Degrees • Aphasia is a neuropsychological disorder that disrupts speech, often resulting from a stroke. After initially acquiring aphasia, patients often experience rapid recovery in much of their language performance (dependent variable) as time (independent variable) increases. But, this recovery eventually slows down, and language performance doesn’t get much better no matter how much more time increases (e.g., Demeurisse et al., 1980). Linear trend: Quadratic trend: POSITIVE NEGATIVE
  • 45. Quadratic & Higher Degrees • Studies of practice and expertise (e.g., Logan, 1988) show that people learning to do a task—such as arithmetic—initially show a quick decrease in response time (dependent variable) as the amount of practice increases. However, eventually they hit the point where the task can’t possibly be done any faster, and response time reaches an asymptote and stops decreasing. Linear trend: Quadratic trend:
  • 46. Quadratic & Higher Degrees • Studies of practice and expertise (e.g., Logan, 1988) show that people learning to do a task—such as arithmetic—initially show a quick decrease in response time (dependent variable) as the amount of practice increases. However, eventually they hit the point where the task can’t possibly be done any faster, and response time reaches an asymptote and stops decreasing. Linear trend: Quadratic trend: NEGATIVE POSITIVE
  • 47. Quadratic & Higher Degrees • Could go up to even higher degrees (Time3, Time4…) • degree=3 if highest exponent is 3 • Degree minus 1 = Number of bends in the curve -100 -50 0 50 100 x^3 0 20 40 60 80 100 x^1 0 20 40 60 80 100 x^2 0 bends 1 bend 2 bends
  • 48. Quadratic & Higher Degrees • Maximum degree of polynomial: # of time points minus 1 • Example: 2 time points perfectly fit by a line (degree 1). Nothing left for a quadratic term to explain. • But, don’t want to overfit • Probably not the case that the real underlying (population) trajectory has 6 bends in it • What degree should we include? • Theoretical considerations • If comparing conditions, look at mean trajectory across conditions (Mirman et al., 2008)
  • 49. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 50. Child 2 Child 1 Child 3 Child 4 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Child 3 Assess- ment 1 Child 3 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 • So far, we assume the same growth rate for all kids • Almost certainly not true! • At level 2, we’re sampling kids both with different starting points (intercepts) and growth rates (slopes) Longitudinal Data: Random Slopes Growth Rate 1 Growth Rate 2
  • 51. Longitudinal Data: Random Slopes RANDOM INTERCEPTS MODEL Kids vary in starting point, but all acquire vocabulary at the same rate over this period WITH RANDOM SLOPES Allows rate of vocab acquisition to vary across kids (as well as intercept)
  • 52. • Let’s allow the MonthsInStudy effect to be different for each Child • model.Slope <- lmer(VocabWords ~ 1 + poly(MonthsInStudy,degree=2,raw=TRUE) + (1 + poly(MonthsInStudy,degree=2,raw=TRUE)|Child), data=vocab.p) Longitudinal Data: Random Slopes Child 2 Child 1 Child 3 Child 4 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Child 3 Assess- ment 1 Child 3 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 Growth Rate 1 Growth Rate 2
  • 53. Longitudinal Data: Random Slopes Substantial variability in the MonthsInStudy slope SD of the slope across children is ~5 words Mean slope is 11 words/mo, but some kids might have a slope of 6 or 16
  • 54. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 55. Time-Variant & -Invariant Variables • We may want to include other (experimental or observational) variables in a GCA model: • e.g., our reading intervention • MonthsInStudy + Reading • Effect of Reading invariant across time • Can only affect the intercept (parallel lines) • MonthsInStudy * Reading • Effect of Reading varies with time • Can affect intercept & slope
  • 56. Time-Variant & -Invariant Variables • Fixed-effect results with the interaction: e.g., Huttenlocher et al., 1991 Also results in faster vocab growth (amplifies + Time effect) Linear growth rate for “No” group: 8.4 words / month Linear growth rate for “Yes” group: 8.4 + 4.7 = 13.1 words / month Effect of reading at time 0 (intercept): ~6 words No effect on curvature (quadratic term)
  • 57. Time-Variant & -Invariant Variables • Can be either: • Time-Invariant Predictor: Same across all time points within a subject • e.g., our intervention, race/ethnicity • Level 2 variables • Time-Varying Predictor: Varies even within a subject, from one time point to another • e.g., hours of sleep, mood • Level-1 variable Child 2 Child 1 Child 1 Assess- ment 1 Child 2 Assess- ment 2 Sampled CHILDREN Sampled TIME POINTS LEVEL 2 LEVEL 1 • Since R automatically figures out what’s a level-1 vs. level-2 variable, we don’t have to do anything special for either kind of variable
  • 58. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab
  • 59. ! Another possibility (in general) is a qualitative change in the trajectory at some point in the range of time observed ! e.g., age of majority, public policy changes, etc. ! Include a breakpoint—a categorical variable that represents whether or not you’re above the point at which the change happens ! mydata %>% mutate(Year2015OrLater= ifelse(Year >= 2015, 1, 0))-> mydata ! New variable is: ! 0 before 2015 ! 1 for 2015 onwards Breakpoints
  • 60. ! Another possibility (in general) is a qualitative change in the trajectory at some point in the range of time observed ! e.g., age of majority, public policy changes, etc. ! Include a breakpoint—a categorical variable that represents whether or not you’re above the point at which the change happens Breakpoints Main effect of breakpoint only – single shift downward (or upward) but same slope Main effect of breakpoint & an interaction – slope also changes
  • 61. ! Can apply this technique to any continuous variables, not just time ! e.g., above/below the poverty line ! As with higher-order polynomials, be wary of overfitting the data ! Most effective if we have an a priori reason to expect a breakpoint somewhere ! Or, use cross-validation to support a more exploratory analysis Breakpoints
  • 62. Week 11.1: Growth Curve Analysis ! Overview ! Reshaping Data ! Growth Curve Analysis ! Time as a Predictor Variable ! Quadratic & Higher Degrees ! Random Slopes ! Time-Variant & -Invariant Variables ! Breakpoints ! Lab