Growth Curve Modeling Day 1 Foundations of Growth

Objectives 1. Describe how latent growth, latent growth trajectory analysis, and growth mixture models

Organization of Class • Discuss the components of model • Simulate data corresponding to

• Free (as in beer and as in speech) • Increasingly used for

• • Integrated Development Environment for R Free (as in beer, not as

Agenda • • Day 1 – Models (means & fancy means) – Linear Trends

Structure of Topics • For each model – Description of model – Simulation example

MODELS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0

A simplified or idealized description or conception of a particular system, situation, or process,

Models • Estimate • Uncertainty This work is licensed under the Creative Commons Attribution-Non.

THE MEAN This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4.

Why do we take the mean? This work is licensed under the Creative Commons

• Take the mean of the following numbers: {1, 4, 7, 9, 8,

As an estimate, the mean minimizes the error This work is licensed under the

THE MEAN AS MODEL This work is licensed under the Creative Commons Attribution-Non. Commercial-Share.

The Mean as Model • Outcome (yi): the value observed on y for datum

Estimates describe the data while the error describes a datum This work is licensed

The Mean as Model Simulation Example 1. Plan the Population – Population size –

The Mean as Model Analysis Example 1. Gather our data 2. Describe our data

QUESTIONS? This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0

LINEAR REGRESSION OF A TREND This work is licensed under the Creative Commons Attribution-Non.

Linear Trend This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4.

Linear Trend Best estimate: ∑(et) = 0 Uncertainty: σt This work is licensed under

Linear regression is nothing more than a fancy mean This work is licensed under

Linear Trend Simulation Example 1. Plan the Population – Number of repeated observations –

Plan Population This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4.

Linear Trend Analysis Example 1. Gather our data 2. Describe our data 3. Analyze

Linear Trend Let’s interpret the results: This work is licensed under the Creative Commons

Nonlinear Change Quadratic change This work is licensed under the Creative Commons Attribution-Non. Commercial-Share.

MULTIPLE LINEAR TRENDS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike

TWO LINEAR TRENDS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike

Multiple Linear Trends Simulation Example 1. Plan the Population – Number of repeated observations

Two Linear Trends New York: Philadelphia: This work is licensed under the Creative Commons

This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International

RANDOM INTERCEPT MODEL This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike

Random Intercepts Analysis Example 1. Gather our data 2. Describe our data 3. Analyze

Interpreting the Results This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike

RANDOM SLOPES This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4.

Random Slopes Simulation Example 1. Plan the Population 2. Conjure the population into existence

RANDOM INTERCEPTS & SLOPES This work is licensed under the Creative Commons Attribution-Non. Commercial-Share.

Random Intercepts & Slopes Simulation Example 1. Plan the Population 2. Conjure the population

Random Intercepts & Slopes Uncreative, straightforward interpretation The average metropolitan area started the past

Meaningful interpretation Home values twelve months ago were distributed a-round a mean of $117

Random Intercepts & Slopes Analysis Example 1. Gather our data 2. Describe our data

Interpreting the results Across the largest 150 metropolitan areas, the median home price per

REVIEW This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0

• • • Models distill important information about the world and have –

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Growth Curve Modeling Day 1 Foundations of Growth Curve Models June 28 & 29, 2018 Michael Bader This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Objectives 1. Describe how latent growth, latent growth trajectory analysis, and growth mixture models are related to, and different from, one another 2. Interpret results all three types of growth models 3. Identify the proper modeling technique for analytical questions regarding change 4. Clearly articulate the benefits, assumptions, and shortcomings of different models of change This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Organization of Class • Discuss the components of model • Simulate data corresponding to the model • Analyze real-life data This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

• Free (as in beer and as in speech) • Increasingly used for research • Excellent graphics through the ggplot library This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

• • Integrated Development Environment for R Free (as in beer, not as in speech) This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Agenda • • Day 1 – Models (means & fancy means) – Linear Trends & Modeling Time – Growth Models Day 2 – Modeling time – Covariates – Latent growth trajectory/growth mixture models This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Structure of Topics • For each model – Description of model – Simulation example – Analytical example (using Zillow) This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

MODELS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

A simplified or idealized description or conception of a particular system, situation, or process, often in mathematical terms, that is put forward as a basis for theoretical or empirical understanding, or for calculations, predictions, etc. ; a conceptual or mental representation of something. Oxford English Dictionary This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Models • Estimate • Uncertainty This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

THE MEAN This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Why do we take the mean? This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

• Take the mean of the following numbers: {1, 4, 7, 9, 8, 2, 5, 3, 10, 6} • Calculate the error for each number This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

The Mean This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

As an estimate, the mean minimizes the error This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

THE MEAN AS MODEL This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

The Mean as Model • Outcome (yi): the value observed on y for datum (observation) i • Estimate (µ): is the best guess (by minimizing the error) of the data as a whole • Error (ei): is the deviation of the mean from the observed outcome for datum (observation) i This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Estimates describe the data while the error describes a datum This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

The Mean as Model Simulation Example 1. Plan the Population – Population size – Set mean – Set standard deviation 2. Conjure the population into existence 3. Analyze the population we created This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

The Mean as Model Analysis Example 1. Gather our data 2. Describe our data 3. Analyze our data 4. Interpret the data This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

QUESTIONS? This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

LINEAR REGRESSION OF A TREND This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear Trend This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear Trend Best estimate: ∑(et) = 0 Uncertainty: σt This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear regression is nothing more than a fancy mean This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear Trend Simulation Example 1. Plan the Population – Number of repeated observations – Initial level of outcome – Change per unit time 2. Conjure the population into existence 3. Analyze the population we created File: 01 b_lineartrend_simulation. R This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Plan Population This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear Trend Analysis Example 1. Gather our data 2. Describe our data 3. Analyze our data 4. Interpret the data File: 01 b_lineartrend_analysis. R This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Linear Trend Let’s interpret the results: This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Nonlinear Change Quadratic change This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

MULTIPLE LINEAR TRENDS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

TWO LINEAR TRENDS This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Multiple Linear Trends Simulation Example 1. Plan the Population – Number of repeated observations – Initial level of outcome (for each city) – Change per unit time (for each city) 2. Conjure the population into existence 3. Analyze the population we created This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Two Linear Trends New York: Philadelphia: This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Multiple Linear Trends This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

RANDOM INTERCEPT MODEL This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercept Model This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Multiple Linear Trends Simulation Example 1. Plan the Population – Number of repeated observations – Initial level of outcome – Variation in initial level of outcome across cities – Change per unit time 2. Conjure the population into existence 3. Analyze the population we created This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercepts Analysis Example 1. Gather our data 2. Describe our data 3. Analyze our data 4. Interpret the data This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Interpreting the Results This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

RANDOM SLOPES This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Slopes This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Slopes Simulation Example 1. Plan the Population 2. Conjure the population into existence 3. Analyze the population we created This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

RANDOM INTERCEPTS & SLOPES This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercepts & Slopes This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercepts & Slopes Simulation Example 1. Plan the Population 2. Conjure the population into existence 3. Analyze the population we created This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercepts & Slopes Uncreative, straightforward interpretation The average metropolitan area started the past year with (logged) home values per square foot of 4. 763, with a standard of 0. 093. (Logged) values increased, on average, by 0. 0048 per month, with a standard deviation of 0. 0000255. This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Meaningful interpretation Home values twelve months ago were distributed a-round a mean of $117 and a standard deviation of 9. 2% in these metropolitan areas. The average metropolitan area saw home values increase by 0. 48% per month, with a standard deviation of 0. 5%. . This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Random Intercepts & Slopes Analysis Example 1. Gather our data 2. Describe our data 3. Analyze our data 4. Interpret the data This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

Interpreting the results Across the largest 150 metropolitan areas, the median home price per square foot in last April was $123, but prices varied by a standard deviation of 46%. Since last April, median home prices increased, on average, by 0. 55% per month, or 6. 6% over the year. Increases in home prices varied around the average increase by 0. 02% percent per month. The modest correlation between the random terms of the intercept and slope means that increases were larger in metropolitan areas that started with higher median values. This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

REVIEW This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.

• • • Models distill important information about the world and have – An outcome – Deterministic component (data-level) – Stochastic component (datum-level) All of our models attempt to minimize error (everything we do creates fancy means) Determine what varies in the model and then analyze it This work is licensed under the Creative Commons Attribution-Non. Commercial-Share. Alike 4. 0 International License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-nc-sa/4. 0/.