UQ Library 2024-09-17
- What are we going to learn?
- Keep in mind
- Open RStudio
- Setting up
- Statistics in R using base and stats
- The inbetween…
- Introducing Tidymodels
- Use a model to predict
- Close project
- Useful links
In this hands-on session, you will use R, RStudio to run analysis of variance (ANOVA) and linear regression models.
Specifically, you will learn:
- visualize data in base R and using the ggplot2 package
- analysis of variance (ANOVA) in base R
- linear models in base R
- the tidy model approach
- Everything we write today will be saved in your project. Please remember to save it in your H drive or USB if you are using a Library computer.
- R is case sensitive: it will tell the difference between uppercase and lowercase.
- Respect the naming rules for objects (no spaces, does not start with a number…)
For any dataset or function doubts that you might have, don’t forget the three ways of getting help in RStudio:
- the shortcut command:
?functionname - the help function:
help(functionname) - the keyboard shortcut: press F1 after writing a function name
- If you are using your own laptop please open RStudio
- If you need them, we have installation instructions
- Make sure you have a working internet connection
- On Library computers (the first time takes about 10 min.):
- Log in with your UQ credentials (student account if you have two)
- Make sure you have a working internet connection
- Go to search at bottom left corner (magnifiying glass)
- Open the ZENworks application
- Look for RStudio
- Double click on RStudio which will install both R and RStudio
# install.packages("readr")
library(readr) # for importing data
# install.packages("dplyr")
library(dplyr) # data manipulation package
# install.packages("ggplot2")
library(ggplot2) # data visualization
# install.packages("car")
library(car) # Companion to Applied Regression package
# install.packages("performance")
library(performance) # Assessment of Regressmon Models performanceRemember to use Ctrl+Enter to execute a command from the script.
- Click the “File” menu button (top left corner), then “New Project”
- Click “New Directory”
- Click “New Project” (“Empty project” if you have an older version of RStudio)
- In “Directory name”, type the name of your project, e.g. “dplyr_intro”
- Select the folder where to locate your project: for example, the
Documents/RProjectsfolder, which you can create if it doesn’t exist yet. - Click the “Create Project” button
We will use a script to write code more comfortably.
- Menu: Top left corner, click the green “plus” symbol, or press the shortcut (for Windows/Linux) Ctrl+Shift+N or (for Mac) Cmd+Shift+N. This will open an “Untitled1” file.
- Go to “File > Save” or press (for Windows/Linux) Ctrl+S or (for Mac) Cmd+S. This will ask where you want to save your file and the name of the new file.
- Call your file “process.R”
The following section will be using data from Constable (1993) to explore how three different feeding regimes affect the size of sea urchins over time.
Sea urchins reportedly regulate their size according to the level of food available to regulate maintenance requirements. The paper examines whether a reduction in suture width (i.e., connection points between plates; see Fig 1 from constable 1993) is the basis for shrinking due to low food conditions.
The data in csv format is available from the tidymodels website and were assembled for a tutorial here.
urchins <-
# read in the data
read_csv("https://tidymodels.org/start/models/urchins.csv") %>%
# change the names to be more description
setNames(c("food_regime", "initial_volume", "width")) %>%
# convert food_regime from chr to a factor, helpful for modeling
mutate(food_regime = factor(food_regime,
levels = c("Initial", "Low", "High")))urchins # see the data as a tibble# A tibble: 72 × 3
food_regime initial_volume width
<fct> <dbl> <dbl>
1 Initial 3.5 0.01
2 Initial 5 0.02
3 Initial 8 0.061
4 Initial 10 0.051
5 Initial 13 0.041
6 Initial 13 0.061
7 Initial 15 0.041
8 Initial 15 0.071
9 Initial 16 0.092
10 Initial 17 0.051
# ℹ 62 more rows
We have 72 urchins with data on:
- experimental feeding regime group with 3 levels (Initial, Low, or High)
- size in milliliters at the start of the experiment (initial_volume)
- suture width in millimeters at the end of the experiment (width, see Fig 1)
Use a boxplot to visualize width versus food_regime as a factor and a
scatterplot for width versus initial_volume as a continuous variable.
boxplot(width ~ food_regime, data = urchins)
plot(width ~ initial_volume, data = urchins)
We can see that there are some relationships between the response variable (width) and our two covariates (food_regime and initial volume). But what about the interaction between the two covariates?
Challenge 1 - Use ggplot2 to make a plot visualizing the interaction between our two variables. Add a trendline to the data.
Hint: think about grouping and coloring.
ggplot(urchins,
aes(x = initial_volume,
y = width,
col = food_regime)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) + # add a linear trendline with out a confidence interval e.g., se = FALSE
scale_color_viridis_d(option = "plasma", end = 0.7) # change to color blind friendly palette, end is a corrected hue value`geom_smooth()` using formula = 'y ~ x'
Urchins that were larger in volume at the start of the experiment tended to have wider sutures at the end. Slopes of the lines look different so this effect may depend on the feeding regime indicating we should include an interaction term.
Information in this section was taken from rpubs.com and Data Analysis in R Ch 7.
We can do an ANOVA with the aov() function to test for differences in
sea urchin suture width between our groups. We are technically running
and analysis of covariance (ANCOVA) as we have both a continuous and
a categorical variable. ANOVAs are for categorical variables and we will
see that some of the post-hoc tests are not amenable to continuous
variables.
aov()uses the model formularesponse variable ~ covariate1 + covariate2. The * denotes the inclusion of both main effects and interactions which we have done below. The formula below is equivalent toreponse ~ covar1 + covar2 + covar1:covar2i.e., the main effect of covar 1 and covar 2, and the interaction between the two.
aov_urch <- aov(width ~ food_regime * initial_volume,
data = urchins)
summary(aov_urch) # print the summary statistics Df Sum Sq Mean Sq F value Pr(>F)
food_regime 2 0.012380 0.006190 13.832 9.62e-06 ***
initial_volume 1 0.008396 0.008396 18.762 5.15e-05 ***
food_regime:initial_volume 2 0.004609 0.002304 5.149 0.00835 **
Residuals 66 0.029536 0.000448
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Both the main effects and interaction are significant (p < 0.05) indicating a significant interactive effect between food regime and initial volume on urchin suture width. We need to do a pairwise-comparison to find out which factor levels and combination of the two covariates have the largest effect on width.
Run a Tukey’s Honestly Significant Difference (HSD) test - note it does not work for non-factors as per the warning message.
TukeyHSD(aov_urch) Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = width ~ food_regime * initial_volume, data = urchins)
$food_regime
diff lwr upr p adj
Low-Initial -0.006791667 -0.021433881 0.007850548 0.5100502
High-Initial 0.023791667 0.009149452 0.038433881 0.0006687
High-Low 0.030583333 0.015941119 0.045225548 0.0000129
The comparison between High-Initial and High-Low food regimes are significant (p < 0.05).
We also want to check that our model is a good fit and does not violate any ANOVA assumptions:
- Data are independent and normally distributed.
- The residuals from the data are normally distributed.
- The variances of the sampled populations are equal.
Challenge 2 - Use a histogram and qqplots to visually check data are normal.
hist(urchins$width)
qqnorm(urchins$width)
qqline(urchins$width)
You could also run a Shapiro-Wilk test on the data:
shapiro.test(urchins$width) Shapiro-Wilk normality test
data: urchins$width
W = 0.95726, p-value = 0.01552
The p-value is less than 0.05 so the data are significantly different from a normal distribution.
Check the model residuals. Plot the residuals vs fitted values - do not want too much deviation from 0.
plot(aov_urch, 1)
Can also plot the predicted values from the model with the acutal values.
plot(predict(aov_urch) ~ urchins$width)
abline(0, 1, col = "red") # plot a red line with intercept of 0 and slope of 1
Check the normality of residuals, run Shapiro-Wilk test on residuals:
plot(aov_urch, 2)
shapiro.test(resid(aov_urch)) Shapiro-Wilk normality test
data: resid(aov_urch)
W = 0.98456, p-value = 0.5244
The residuals fall on the Normal Q-Q plot diagonal and the Shapiro-Wilk result is non-significant (p > 0.05).
Check for homogeneity of variance
Challenge 3 - use the help documentation for leveneTest() from the
car package to check homogenetity of variance on food_regime.
Again, only works for factor groups.
leveneTest(width ~ food_regime, data = urchins)Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 2 4.4224 0.01559 *
69
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The Levene’s Test is significant for food_regime (not what we want)
and there are a few options to deal with this. You can ignore this
violation based on your own a priori knowledge of the distribution of
the population being samples, drop the p-value significance, or use a
different test.
lm_urch <- lm(width ~ food_regime * initial_volume,
data = urchins)
summary(lm_urch)Call:
lm(formula = width ~ food_regime * initial_volume, data = urchins)
Residuals:
Min 1Q Median 3Q Max
-0.045133 -0.013639 0.001111 0.013226 0.067907
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0331216 0.0096186 3.443 0.001002 **
food_regimeLow 0.0197824 0.0129883 1.523 0.132514
food_regimeHigh 0.0214111 0.0145318 1.473 0.145397
initial_volume 0.0015546 0.0003978 3.908 0.000222 ***
food_regimeLow:initial_volume -0.0012594 0.0005102 -2.469 0.016164 *
food_regimeHigh:initial_volume 0.0005254 0.0007020 0.748 0.456836
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02115 on 66 degrees of freedom
Multiple R-squared: 0.4622, Adjusted R-squared: 0.4215
F-statistic: 11.35 on 5 and 66 DF, p-value: 6.424e-08
In the output, we have the model call, residuals, and the coefficients.
The first coefficient is the (Intercept) and you might notice the
food_regimeInitial is missing. The function defaults to an effects
parameterization where the intercept is the reference or baseline of the
categorical group - Initial in this case.
You can change the reference level of a factor using the
relevel()function.
The estimates of the remaining group levels of food_regime represents
the effect of being in that group. To calculate the group coefficients
for all group levels you add the estimates for the level to the
intercept (first group level) estimate. For example, the estimate for
the ‘Initial’ feeding regime is 0.0331 and we add the estimate of ‘Low’
(0.0331 + 0.0197) to get the mean maximum size of 0.0528 mm for width.
For the continuous covariate, the estimate represents the change in the response variable for a unit increase in the covariate. ‘Initial Volume’s’ estimate of 0.0015 represents a 0.0015 mm increase (the estimate is positive) in width per ml increase in urchin initial volume.
We can get ANOVA test statistics on our linear model using the anova()
in base or Anova() from the car package.
anova(lm_urch)Analysis of Variance Table
Response: width
Df Sum Sq Mean Sq F value Pr(>F)
food_regime 2 0.0123801 0.0061900 13.8321 9.616e-06 ***
initial_volume 1 0.0083962 0.0083962 18.7621 5.154e-05 ***
food_regime:initial_volume 2 0.0046088 0.0023044 5.1494 0.008354 **
Residuals 66 0.0295358 0.0004475
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(lm_urch)Anova Table (Type II tests)
Response: width
Sum Sq Df F value Pr(>F)
food_regime 0.0168653 2 18.8434 3.358e-07 ***
initial_volume 0.0083962 1 18.7621 5.154e-05 ***
food_regime:initial_volume 0.0046088 2 5.1494 0.008354 **
Residuals 0.0295358 66
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These are effectively the same as the aov() model we ran before.
Note: The statistics outputs are the same comparing the
aov()andanova()models while theAnova()model is not exactly the same. TheAnova()output tells us it was a Type II test and theaov()documentation says it is only for balanced designs which means the Type 1 test is the applied (see here). The type of test can be set forAnova()but not the others. Here, the overall take-away from the different ANOVA functions are comparable.
Challenge 4 - use the check_model() documentation to apply the
function to our lm_urch model.
The performance package has a handy function check_model() that will
check several aspects of your model in one go:
check_model(lm_urch)
Challenge 5 - conduct your own ANOVA or linear regression using the mgp dataset from {ggplot2}.
- Test whether # of cylinders and/or engine displacement affect fuel efficiency.
- Make a plot to visualize the relationship.
Hint: Check out the documentation for the dataset
?mpgto see the varaibles in the dataset. Are the variables the right data type? Suggest saving the dataset locally in your environment i.e.,mpg2 <- mpgso you can change data types if necessary.
mpg2 <- mpg
mpg2$cyl <- as.factor(mpg$cyl) # convert cyl from numeric to factor
# base R ANOVA
aov_cars <- aov(hwy ~ cyl * displ, data = mpg2)
summary(aov_cars) Df Sum Sq Mean Sq F value Pr(>F)
cyl 3 4836 1612.1 142.69 < 2e-16 ***
displ 1 219 218.8 19.37 1.66e-05 ***
cyl:displ 2 642 321.1 28.42 9.65e-12 ***
Residuals 227 2565 11.3
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
TukeyHSD(aov_cars, "cyl") Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = hwy ~ cyl * displ, data = mpg2)
$cyl
diff lwr upr p adj
5-4 -0.05246914 -4.508182 4.403244 0.9999898
6-4 -5.97968433 -7.355259 -4.604110 0.0000000
8-4 -11.17389771 -12.593534 -9.754261 0.0000000
6-5 -5.92721519 -10.385582 -1.468848 0.0038274
8-5 -11.12142857 -15.593586 -6.649271 0.0000000
8-6 -5.19421338 -6.622156 -3.766270 0.0000000
# linear model
lm_cars <- lm(hwy ~ cyl * displ, data = mpg2)
summary(lm_cars)Call:
lm(formula = hwy ~ cyl * displ, data = mpg2)
Residuals:
Min 1Q Median 3Q Max
-6.6698 -2.0533 -0.4563 1.6948 13.1597
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 46.601 2.586 18.018 < 2e-16 ***
cyl5 2.887 1.773 1.628 0.10484
cyl6 -10.221 3.795 -2.693 0.00760 **
cyl8 -35.626 4.390 -8.114 3.07e-14 ***
displ -8.295 1.193 -6.954 3.74e-11 ***
cyl5:displ NA NA NA NA
cyl6:displ 4.318 1.440 2.998 0.00302 **
cyl8:displ 9.591 1.376 6.969 3.43e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.361 on 227 degrees of freedom
Multiple R-squared: 0.6896, Adjusted R-squared: 0.6814
F-statistic: 84.05 on 6 and 227 DF, p-value: < 2.2e-16
Anova(lm_cars) # from the car packageNote: model has aliased coefficients
sums of squares computed by model comparison
Anova Table (Type II tests)
Response: hwy
Sum Sq Df F value Pr(>F)
cyl 207.16 3 6.1123 0.0005139 ***
displ 218.78 1 19.3655 1.661e-05 ***
cyl:displ 642.17 2 28.4211 9.654e-12 ***
Residuals 2564.50 227
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
ggplot(data = mpg2,
aes(x = displ,
y = hwy,
color = cyl)) +
geom_point() +
geom_smooth(method = "lm") `geom_smooth()` using formula = 'y ~ x'
Before going into Tidymodels, it should be mentioned there are many excellent linear regression packages. To name a few:
- nlme
- lmer
- lmerTest
- glmmTMB
- and more…
The packages vary in the methods, how to specify random factors, etc. The model outputs also tend to be not so friendly to export into a table and document.
Like the tidyverse package, the Tidymodels framework is a collection of packages for modeling and machine learning following the tidyverse principles.
This section is modified from the first Tidymodels article.
# install.packages("tidymodels")
library(tidymodels) # for parsnip package and rest of tidymodels
# install.packages("dotwhisker")
# dotwhiskers was archived from CRAN in 2024, see:
# https://github.com/fsolt/dotwhisker/issues/115
# if unavailable, can be installed from GitHub:
# remotes::install_github('fsolt/dotwhisker')
library(dotwhisker)# for visualizing regression results
# install.packages("parsnip")
library(parsnip)Let’s apply a standard two-way analysis of variance (ANOVA) model to the dataset as we did before. For this kind of model, ordinary least squares is a good initial approach.
For Tidymodels, we need to specify the following:
- The functional form using the parsnip package.
- The method for fitting the model by setting the engine.
We will specify the functional form or model type as “linear regression” as there is a numeric outcome with a linear slope and intercept. We can do this with:
linear_reg() Linear Regression Model Specification (regression)
Computational engine: lm
On its own, not that interesting. Next, we specify the method for
fitting or training the model using the set_engine() function. The
engine value is often a mash-up of the software that can be used to fit
or train the model as well as the estimation method. For example, to use
ordinary least squares, we can set the engine to be lm.
The documentation page for linear_reg() lists the possible engines. We’ll save this model object as lm_mod.
lm_mod <-
linear_reg() %>%
set_engine("lm")Next, the model can be estimated or trained using the fit() function
and the model formula we used for the ANOVA:
width ~ initial_volume * food_regime
lm_fit <-
lm_mod %>%
fit(width ~ initial_volume * food_regime, data = urchins)
lm_fitparsnip model object
Call:
stats::lm(formula = width ~ initial_volume * food_regime, data = data)
Coefficients:
(Intercept) initial_volume
0.0331216 0.0015546
food_regimeLow food_regimeHigh
0.0197824 0.0214111
initial_volume:food_regimeLow initial_volume:food_regimeHigh
-0.0012594 0.0005254
We can use the tidy() function for our lm object to output model
parameter estimates and their statistical properties. Similar to
summary() but the results are more predictable and useful format.
tidy(lm_fit)# A tibble: 6 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 0.0331 0.00962 3.44 0.00100
2 initial_volume 0.00155 0.000398 3.91 0.000222
3 food_regimeLow 0.0198 0.0130 1.52 0.133
4 food_regimeHigh 0.0214 0.0145 1.47 0.145
5 initial_volume:food_regimeLow -0.00126 0.000510 -2.47 0.0162
6 initial_volume:food_regimeHigh 0.000525 0.000702 0.748 0.457
This output can be used to generate a dot-and-whisker plot of our
regression results using the dotwhisker package:
tidy(lm_fit) %>%
dwplot(dot_args = list(size = 2, color = "black"),
whisker_args = list(color = "black"),
vline = geom_vline(xintercept = 0,
color = "grey50",
linetype = 2))
Say that it would be interesting to make a plot of the mean body size for urchins that started the experiment with an initial volume of 20 ml.
First, lets make some new example data to predict for our graph:
new_points <- expand.grid(initial_volume = 20,
food_regime = c("Initial", "Low", "High"))
new_points initial_volume food_regime
1 20 Initial
2 20 Low
3 20 High
We can then use the predict() function to find the mean values at 20
ml initial volume.
With tidymodels, the types of predicted values are standardized so that we can use the same syntax to get these values.
Let’s generate the mean suture width values:
mean_pred <- predict(lm_fit, new_data = new_points)
mean_pred# A tibble: 3 × 1
.pred
<dbl>
1 0.0642
2 0.0588
3 0.0961
When making predictions, the tidymodels convention is to always produce a tibble of results with standardized column names. This makes it easy to combine the original data and the predictions in a usable format:
conf_int_pred <- predict(lm_fit,
new_data = new_points,
type = "conf_int")
conf_int_pred# A tibble: 3 × 2
.pred_lower .pred_upper
<dbl> <dbl>
1 0.0555 0.0729
2 0.0499 0.0678
3 0.0870 0.105
# now combine:
plot_data <-
new_points %>%
bind_cols(mean_pred, conf_int_pred)
plot_data initial_volume food_regime .pred .pred_lower .pred_upper
1 20 Initial 0.06421443 0.05549934 0.07292952
2 20 Low 0.05880940 0.04986251 0.06775629
3 20 High 0.09613343 0.08696233 0.10530453
# and plot:
ggplot(plot_data,
aes(x = food_regime)) +
geom_point(aes(y = .pred)) +
geom_errorbar(aes(ymin = .pred_lower,
ymax = .pred_upper),
width = .2) +
labs(y = "urchin size")
There is also an example of a Bayesian model in the tidymodels article I have not included here.
Closing RStudio will ask you if you want to save your workspace and scripts. Saving your workspace is usually not recommended if you have all the necessary commands in your script.
- For statistical analysis in R:
- Steve Midway’s Data Analysis in R Part II Analysis
- Jeffrey A. Walker’s Applied Statistics for Experiemental Biology
- Chester Ismay and Albert Y. Kim’s ModernDive Statistical Inference via Data Science
- For tidymodels:
- Our compilation of general R resources