Data Types and Subsetting in R

Data Table Subsetting

• Data table subsetting structure: DT[i, j, by], where DT is the data table, i is the rows, j is the columns, and by is the grouping variable

Data Types in R

• Logical: used for logical data (TRUE or FALSE)
• Integer: used for integer type data (whole numbers like 0, 1, 2)
• Numeric: used for real numbers (1.1, 4.8) and can be used for integer data (less efficient)
• Factor: special representation of numeric data when data are fundamentally discrete (e.g., study condition coded as 0 = control, 1 = medication, 2 = psychotherapy)
• Characters: used for text type data (names, qualitative data, etc.) and can store numbers as strings

Operators

• Logical operators: used to manage data (e.g., find outliers, values greater/less than a score)
• Boolean values: TRUE or FALSE, where TRUE is treated as 1 and FALSE is treated as 0 in arithmetic
• Operators:
  • = OR %ge%: Greater than or equal
  • %gl%: Greater than AND less than
  • %gel%: Greater than or equal AND less than
  • %gle%: Greater than AND less than or equal
  • %gele%: Greater than or equal AND less than or equal
  • %in%: In
  • %!in% or %nin%: Not in
  • %c%: Chain operations on the RHS together
  • %e%: Set operator, to use set notation

Subsetting Data

• Subsetting: excluding outliers, selecting participants who meet certain criteria
• Order of subsetting matters

Merging Data

• Rules: one join at a time, x dataset is always on the left, y dataset is always on the right
• Types of joins:
  • Natural join: resulting data has only rows present in both x and y (all = FALSE)
  • Full outer join: resulting data has all rows in x and all rows in y (all = TRUE)
  • Left outer join: resulting data has all rows in x (all.x = TRUE)
  • Right outer join: resulting data has all rows in y (all.y = TRUE)

Reshaping Data

• Necessary for repeated measures/longitudinal/panel data
• Types of data structures:
  • Wide: each measure has a separately-named variable for each time point it was measured
    • Each entity occupies their own row, and each variable occupies a single column
    • Easy to read and interpret, used in descriptive statistics and reporting
  • Long: time point (or wave) is a variable, IDs will have multiple rows
    • Machine-friendly data structure, easier to perform functions like filtering and aggregating
    • Easier to add new data and avoids the problem of null values

Scoring Questionnaire Scales

• There are two ways to score questionnaire scales: adding together to get a sum total score, or taking an average of all items
• Using rowMeans() allows you to perform calculations excluding missing data, which is commonly done and sensible when dealing with small amounts of missing data
• If you want to deal with missing data but need a total score, you can use rowMeans() and multiply the results by the number of items that should have been completed

Cronbach's Alpha

• psych::alpha() is the function to find Cronbach's alpha, a common measure of scale reliability
• psych:: is used to specify the alpha function from the psych package, as alpha is a popular function name

Bivariate Plots

• A bivariate plot shows the relationship between two variables, mapped onto the x- and y-axes
• geom_point() is used to make scatterplots, and additional arguments can be added using + (e.g., geom_line())
• geom_bar() is used for barplots

Best Practices in Data Visualization

• Data to ink ratio: aim for more data and less ink
• Themes are helpful in achieving this goal
• Axes can be useful for providing more data, such as labeling with quantiles
• Shapes on scatterplots help identify categorical variables

Types of Plots

Violin Plots

• Used to compare the distribution of data between groups
• Thicker regions have more points, narrow regions have fewer data points
• Show the range/spread of each variable and mean and confidence interval summaries

Histograms

• Define equal width bins on the x-axis and count how many observations fall within each bin
• Bars display these, where the width of the bar is the width of the bin and the height is the count (frequency) of observations
• Show a univariate distribution

Density Plots

• Show the distribution using a smooth density function rather than binning data
• Height indicates the relative frequency of observations at a particular value
• Designed so that they sum to one
• Show a univariate distribution

Dot Plots

• Effective at showing raw data for small datasets
• Each dot represents one person
• Dots are stacked on top of each other if they would overlap
• Provide greater precision than histograms
• Show a univariate distribution

QQ Plots

• A scatterplot created by plotting two sets of quantiles against one another
• If both sets of quantiles came from the same distribution, the points should form a line that's roughly straight

Scoring Questionnaire Scales

• There are two ways to score questionnaire scales: adding together to get a sum total score, or taking an average of all items
• Using rowMeans() allows you to perform calculations excluding missing data, which is commonly done and sensible when dealing with small amounts of missing data
• If you want to deal with missing data but need a total score, you can use rowMeans() and multiply the results by the number of items that should have been completed

Cronbach's Alpha

• psych::alpha() is the function to find Cronbach's alpha, a common measure of scale reliability
• psych:: is used to specify the alpha function from the psych package, as alpha is a popular function name

Bivariate Plots

• A bivariate plot shows the relationship between two variables, mapped onto the x- and y-axes
• geom_point() is used to make scatterplots, and additional arguments can be added using + (e.g., geom_line())
• geom_bar() is used for barplots

Best Practices in Data Visualization

• Data to ink ratio: aim for more data and less ink
• Themes are helpful in achieving this goal
• Axes can be useful for providing more data, such as labeling with quantiles
• Shapes on scatterplots help identify categorical variables

Types of Plots

Violin Plots

• Used to compare the distribution of data between groups
• Thicker regions have more points, narrow regions have fewer data points
• Show the range/spread of each variable and mean and confidence interval summaries

Histograms

• Define equal width bins on the x-axis and count how many observations fall within each bin
• Bars display these, where the width of the bar is the width of the bin and the height is the count (frequency) of observations
• Show a univariate distribution

Density Plots

• Show the distribution using a smooth density function rather than binning data
• Height indicates the relative frequency of observations at a particular value
• Designed so that they sum to one
• Show a univariate distribution

Dot Plots

• Effective at showing raw data for small datasets
• Each dot represents one person
• Dots are stacked on top of each other if they would overlap
• Provide greater precision than histograms
• Show a univariate distribution

QQ Plots

• A scatterplot created by plotting two sets of quantiles against one another
• If both sets of quantiles came from the same distribution, the points should form a line that's roughly straight

Simple vs. Multiple Linear Regression

• Simple linear regression: equation yi = b0 + b1 * xi + εi, where yi is the outcome variable, xi is the predictor/explanatory variable, εi is the residual/error term, b0 is the intercept, and b1 is the slope of the line.
• In simple linear regression, the model parameters (b0 and b1) are the same for all participants, but each person has their own values of yi and xi, and there will be some unexplained residual (εi).
• If we want to talk about only what is predicted based on the regression coefficients, we can write yi = b0 + b1 * xi, leaving off the residual error term (εi).

Multiple Linear Regression

• Multiple linear regression works in principle basically the same way as simple linear regression, but allows for more than one predictor (explanatory) variable in a single model.
• The equation for multiple linear regression is yi = b0 + b1 * x1i + ... + bk * xki + εi, where yi is the outcome variable, x1i, x2i, ..., xki are the predictor variables, and εi is the residual/error term.
• The regression coefficients (b0, b1, ..., bk) are interpreted fairly similarly to those in simple linear regression, but with some extra requirements.
• b0 is the intercept, the expected (model predicted) value of yi when all predictors are 0.
• b1, b2, ..., bk are the slopes of the line, capturing how much yi is expected to change for a one unit change in x1, x2, ..., xk, respectively, holding all other predictors constant.

Line of Best Fit and Residuals

• The line of best fit is the regression line that goes through the data points, minimizing all the residuals.
• The residuals are the differences between the model predicted values and the observed values.

Generalized Linear Models (GLMs)

• GLMs extend the linear model to different outcomes, such as continuous, normally distributed variables (linear regression), binary 0/1 variables (logistic regression, probit regression), and count variables (poisson regression, negative binomial regression).
• GLMs force things to be linear, using some function to link or transform eta (n).
• In linear regression, there is no link function because it's already in linear space.

Probability Distribution

• A probability distribution is a function where, as we move along the x-axis, the y-axis is telling us what is the probability of that value occurring.

Normal Distribution (Gaussian Distribution)

• Parameters: mean and standard deviation.

R Output from a Linear Model

• The lm() function in R is used to fit a linear model, and it uses a formula interface to specify the desired model, with the format outcome ~ predictor.
• The summary() function provides a quick summary of the model, including the regression coefficients, standard errors, t-values, and p-values.
• The confint() function provides confidence intervals for the regression coefficients.

Linear Regression Assumptions

• L.I.N.E. = there is a linear relationship, variables and errors are independent, errors are normally distributed, and there needs to be equal variance.
• Independent variables: all values of the outcome should come from a different person.
• Errors: for any pair of observations, the error terms should be uncorrelated.
• Normally-distributed errors: the errors (i.e., the residuals) should be random and normally distributed with a mean of 0.
• Equal variance/homoscedasticity: for each value of the predictors, the variance of the error term should be constant.

Model Diagnostics

• To assess normally-distributed errors, look at the density plot of residuals (black line) vs a normal distribution (dotted blue line).
• To identify outliers, look at the QQ plot of residuals (black points are outliers).
• To check for equal variance/homoscedasticity, look at a scatterplot of the model predicted values against the residuals - basically, we want the residuals to be about the same as the predicted values (i.e., blue dotted lines to be horizontal and parallel).

Poisson Regression

• Used for count variables, which are discrete and must be whole, 0 or positive numbers
• Poisson distribution is used when counts are relatively rare
• Examples of research use cases:
  • Examining risk factors for accidents over a 12-month period
  • Analyzing the number of children people have
  • Predicting how many friends people have
  • Evaluating whether an intervention reduced medication non-adherence
  • Testing whether treating mental health can lower healthcare appointments
• Poisson regression:
  • Does not assume normal distribution
  • Has one parameter, lambda (mean and variance)
  • Linear regression rarely works well for count outcomes due to:
    • Straight line being a bad fit at extremes
    • Non-normal distribution of residuals
• General linear regression deals with Poisson regression using:
  • Link functions to transform linear predicted values to never go below 0
  • Assuming a Poisson distribution
  • Defining the link function as: η = g(λ) = ln(λ)
• Assumptions of Poisson regression:
  • Poisson distribution, outcome is counts, positive integers
  • Mean and variance must be the same
  • Linear relationship on the link scale (ln)
  • No need to worry about normally distributed errors or equal variance/homoscedasticity
  • Watch for right-side outliers (extremely high counts)
  • Importance of large sample size
• How to do Poisson regression in R:
  • Use the glm() function with the argument 'family = poisson'
• Incident Rate Ratios (IRRs):
  • Interpret as: "for each one unit higher predictor score, there are IRR times as many events of the outcome"
  • Example: IRR = 2, base rate = 1, one unit higher would be 1*2 = 2

Binary Logistic Regression

• Used for binary outcomes, where the outcome only takes on two values: 0 or 1
• Examples of research use cases:
  • Predicting whether someone will have major depression or not
  • Determining the probability of patients remitting from major depression
  • Predicting the probability of readmission to the hospital within 30 days
  • Predicting the probability of death before age 60
• Binary logistic regression:
  • Linear regression will not work for binary outcomes due to:
    • Straight line being a bad fit
    • Non-normal distribution of residuals
• General linear regression deals with binary logistic regression using:
  • Link functions to transform linear predicted values to never go below 0 and never go above 1
  • Assuming a Bernoulli distribution
  • Defining the link function as: η = g(μ) = ln(μ/1−μ)
• Assumptions of logistic regression:
  • Bernoulli distribution, outcome is probability of event occurring
  • Linear relationship on the link scale (ln)
  • Independent variables, independent errors
  • Identify outliers/extreme values on the predictors
  • Check for separation (predictor variable perfectly predicts the outcome)
  • Importance of large sample size
• How to do logistic regression in R:
  • Use the glm() function with the argument 'family = binomial'
• Odds ratio:
  • Indicates how many more times the odds of the outcome occurring will be for a one unit change in the predictor
  • Higher than 1 means a positive relationship, less than 1 means a negative relationship
• Marginal effect:
  • Instantaneous effect of change at a particular point
  • Equivalent to the slope of a straight line at that value

Missing Data

• Missing data are common and problematic, leading to biased results and efficiency loss.

Types of Missing Data

• Missing completely at random (MCAR): when the missingness mechanism is completely independent of the estimate of our parameter(s) of interest.
• Missing at random (MAR): when the missingness mechanism is conditionally independent of the estimate of our parameter(s) of interest.
• Missing not at random (MNAR): when the missingness mechanism is associated with the estimate of our parameter(s) of interest.

Consequences of Missing Data

• Listwise deletion may lead to biased results unless the data are missing completely at random (MCAR).
• When data are missing completely at random (MCAR), listwise deletion will yield unbiased estimates of the true parameter(s) if the data had not been missing.
• When data are missing at random (MAR), it is possible to recover unbiased estimates if the right other variables are present.

Multiple Imputation (MI)

• Multiple imputation is a robust way to address missing data, involving generating multiple, different datasets with plausible values imputed for the missing data.
• Steps in MI:
• Start with the incomplete data.
• Generate 𝑚 datasets with no missingness, by filling in different plausible values for any missing data.
• Perform the analysis of interest on each imputed dataset.
• Pool the results from the analyses run on each imputed dataset to generate an overall estimate, 𝑄¯.
• Formula to determine total uncertainty in average estimate: T = V¯ + B + B/m, where 𝑉¯ is the average uncertainty estimate of Q̂ across the multiply imputed datasets, B captures the variance in the estimates, Q̂, and m is the number of imputed datasets.

Issues with Using Imputed Datasets

• Issues with using imputed datasets with general linear models:
• Small sample sizes (i.e., 100 or less).
• Colinear variables.
• Lots of interactions between variables.
• Non-normal residuals.

Examining Missing Data

• Before doing any imputation, it is a good idea to examine the data using the VIM package in R.
• The aggr() function shows the proportion of missing data on each individual variable and the patterns of missing data.
• Margin plots can help identify if imputations fall outside the range of observed data or fit with the rest of the trend from the observed data.

Missing Data

• Missing data are common and problematic, leading to biased results and efficiency loss.

Types of Missing Data

• Missing completely at random (MCAR): when the missingness mechanism is completely independent of the estimate of our parameter(s) of interest.
• Missing at random (MAR): when the missingness mechanism is conditionally independent of the estimate of our parameter(s) of interest.
• Missing not at random (MNAR): when the missingness mechanism is associated with the estimate of our parameter(s) of interest.

Consequences of Missing Data

• Listwise deletion may lead to biased results unless the data are missing completely at random (MCAR).
• When data are missing completely at random (MCAR), listwise deletion will yield unbiased estimates of the true parameter(s) if the data had not been missing.
• When data are missing at random (MAR), it is possible to recover unbiased estimates if the right other variables are present.

Multiple Imputation (MI)

• Multiple imputation is a robust way to address missing data, involving generating multiple, different datasets with plausible values imputed for the missing data.
• Steps in MI:
• Start with the incomplete data.
• Generate 𝑚 datasets with no missingness, by filling in different plausible values for any missing data.
• Perform the analysis of interest on each imputed dataset.
• Pool the results from the analyses run on each imputed dataset to generate an overall estimate, 𝑄¯.
• Formula to determine total uncertainty in average estimate: T = V¯ + B + B/m, where 𝑉¯ is the average uncertainty estimate of Q̂ across the multiply imputed datasets, B captures the variance in the estimates, Q̂, and m is the number of imputed datasets.

Issues with Using Imputed Datasets

• Issues with using imputed datasets with general linear models:
• Small sample sizes (i.e., 100 or less).
• Colinear variables.
• Lots of interactions between variables.
• Non-normal residuals.

Examining Missing Data

• Before doing any imputation, it is a good idea to examine the data using the VIM package in R.
• The aggr() function shows the proportion of missing data on each individual variable and the patterns of missing data.
• Margin plots can help identify if imputations fall outside the range of observed data or fit with the rest of the trend from the observed data.

Independence of Observations

• Observations are not always independent, e.g., in longitudinal studies, repeated measures experiments, and clustered data
• This type of data poses challenges to statistical analysis, but can be addressed using linear mixed models

Linear Mixed Models

• Relax the assumption of independence of observations in linear regression
• Allow for variation in observations, including continuous time, missing data, and continuous predictors

Fixed Effects vs Random Effects

• Fixed effects: assume same slope and intercept for all participants, only applicable for one observation per participant
• Random effects: allow for different coefficients (slopes and intercepts) per participant, applicable for repeated measures

Fixed Effects Approximation

• Approximate distribution by: 𝑀 = estimated mean; 𝑆𝐷 = 0 (standard deviation is fixed at 0)

Random Effects Approximation

• Approximate distribution by: 𝑀 = estimated mean; 𝑆𝐷 = estimated standard deviation (SD is free to vary)

Main Difference Between Fixed and Random Effects

• Fixed effects: regression coefficients are the same for everyone
• Random effects: regression coefficients vary randomly for each participant

Intraclass Correlation Coefficient (ICC)

• Measures the ratio of between variance to total variance (ranges between 0 and 1)
• ICC > 0 indicates individual means differ, and individual differences need to be accounted for in analysis

Meandeviations() Function

• Used to calculate between and within versions of a repeated measures variable

Linear Mixed Model Assumptions

• Assume individual units' deviations from the fixed effect follow a normal distribution with mean 0 and standard deviation
• Assume random effect intercept also follows a normal distribution
• Only one additional parameter is needed compared to regular linear regression