16.2 – Causation and Partial correlation

Introduction

Science driven by statistical inference and model building is largely motivated by the the drive to identify pathways of cause and effect linking events and phenomena observed all around us. (We first defined cause and effect in Chapter 2.4) The history of philosophy, from the works of Ancient Greece, China, Middle East and so on is rich in the language of cause and effect. From these traditions we have a number of ways to think of cause and effect, but for us it will be enough to review the logical distinction among three kinds of cause-effect associations:

• Necessary cause
• Sufficient cause
• Contributory cause

Here’s how the logic works. If A is a necessary cause of B, then the mere fact that B is present implies that A must also be present. However, that the presence of A does not imply that B will occur. If A is a sufficient cause of B, then the presence of A necessarily implies the presence of B. However, another cause C may alternatively cause B. Enter the contributory or related cause: A cause may be contributory if the presumed cause A (1) occurs before the effect B, and (2) changing A also changes B. A contributory cause does not need to be necessary nor must it be sufficient; contributory causes play a role in cause and effect.

Thus, following this long tradition of thinking about causality we have the mantra, “Correlation does not imply causation.” The exact phrase was written as early as emphasized by Karl Pearson who invented the statistic back in the late 1800s. This well-worn slogan deserves to be on T-shirts and bumper stickers*, and perhaps to be viewed as the single most important concept you can take from a course in philosophy/statistics. But in practice, we will always be tempted to stray from this guidance. The developments in genome-wide-association studies, or GWAS, are designed to look for correlations, as evidenced by statistical linkage analysis, between variation at one DNA base pair and presence/absence of disease or condition in humans and animal models. These are costly studies to do and in the end, the results are just that, evidence of associations (correlations), not proof of genetic cause and effect. We are less likely to be swayed by a correlation that is weak, but what about correlations that are large, even close to one? Is not the implication of high, statistically significant correlation evidence of causation? No, necessary, but not sufficient.

Spurious correlations

Correlation estimates should be viewed as hypotheses in the scientific sense of the meaning of hypotheses for putative cause-effect pairings. To drive the point home, explore the web site “Spurious Correlations” at https://www.tylervigen.com/spurious-correlations , which allows you to generate X-Y plots and estimate correlations among many different variables. Some of my favorite correlations from “Spurious Correlations” include (Table 1):

Table 1. Spurious correlations, https://www.tylervigen.com/spurious-correlations

These are some pretty strong correlations (cf. effect size discussion, Ch11.4 and 16.1), about as close to +1 as you can get. But really, do you think the amount of cheese that is consumed in the USA has anything to do with the number of PhD degrees awarded in engineering or that apple prices are largely set by the number of lawyers in the USA? Cause and effect implies there must also be some plausible mechanism, not just a strong correlation.

But that does NOT ALSO mean that a high correlation is meaningless. The primary reason a correlation cannot tell about causation is because of the problem (potentially) of an UNMEASURED variable (a confounding variable) being the real driving force (Fig 1).

Figure 1. Unmeasured confounding variables influence association between independent and dependent variables, the characters or traits we are interested in.

Here’s a plot of running times for the best, fastest men and women runners for the 100-meter sprint. Running times over 100 meters of top athletes since the 1920s. The data are collated for you and presented at end of this page (scroll or click here).

Here’s a scatterplot (Fig 2).

Figure 2. Running times over 100 meters of top athletes since the 1920s.

There’s clearly a negative correlation between years and running times. Is the rate of improvement in running times the same for men and women? Is the improvement linear? What, if any, are the possible confounding variables? Height? weight? Biomechanical differences? Society? Training? Genetics? … Performance enhancing drugs…?

If we measure potential confounding factors, we may be able to determine the strength of correlation between two variables that share variation with a third variable.

The partial correlation

There are several ways to work this problem. The partial correlation is a useful way to handle this problem, i.e., where a measured third variable is positively correlated with the two variable you are interested in.

Without formal mathematical proof presented, r_12.3 is the correlation between variables 1 and 2 INDEPENDENT of any covariation with variable 3.

For our running data set, we have the correlation between women’s time for 100m over 9 decades, (r₁₃ = -0.876), between men’s time for 100m over 9 decades (r₂₃ = -0.952), and finally, the correlation we’re interested in, whether men’s and women’s times are correlated (r₁₂ = +0.71). When we use the partial correlation, however, I get r_12.3 = -0.819… much less than 0 and significantly different from zero. In other words, men’s and women’s times are not positively correlated independent of the correlation both share with the passage of time (decades)! The interpretation is that men are getting faster at a rate faster than women.

In conclusion, keep your head about you when you are doing analyses. You may not have the skills or knowledge to handle some problems (partial correlation), but you can think simply — why are two variables correlated? One causes the other to increase (or decrease) OR the two are both correlated with another variable.

Testing the partial correlation

Like our simple correlation, the partial correlation may be tested by a t-test, although modified to account for the number of pairwise correlations (Wetzels and Wagenmakers 2012). The equation for the t test statistic is now

with k equal to the number of pairwise correlations and n – 2 – k degrees of freedom.

Examples

Lead exposure and low birth weigh. The data set is numbers of low birth weight births (< 2,500 g regardless of gestational age) and numbers of children with high levels of lead (10 or more micrograms of lead in a deciliter of blood) measured from their blood. Data used for 42 cities and towns of Rhode Island, United States of America (data at end of this page, scroll or click here to access the data).

A scatterplot of number of children with high lead is shown below (Fig 3).

Figure 3. Scatterplot birth weight by lead exposure.

The product moment correlation was r = 0.961, t = 21.862, df = 40, p < 2.2e-16. So, at first blush looking at the scatterplot and the correlation coefficient, we conclude that there is a significant relationship between lead and low birth weight, right?

However, by the description of the data you should recall that counts were reported, not rates (eg, per 100,000 people). Clearly, population size varies among the cities and towns of Rhode Island. West Greenwich had 5085 people whereas Providence had 173,618. We should suspect that there is also a positive correlation between number of children born with low birth weight and numbers of children with high levels of lead. Indeed there are.

Correlation between Low Birth Weight and Population, r = 0.982

Correlation between High Lead levels and Population, r = 0.891

The question becomes, after removing the covariation with population size is there a linear association between high lead and low birth weight? One option is to calculate the partial correlation. To get partial correlations in Rcmdr, select

Statistics → Summaries → Correlation matrix

then select “partial” and select all three variables (Ctrl key) (Fig 4)

Figure 4. Screenshot Rcmdr partial correlation menu

Results are shown below.

partial.cor(leadBirthWeight[,c("HiLead","LowBirthWeight","Population")], tests=TRUE, use="complete")

Partial correlations:
                 HiLead LowBirthWeight Population
HiLead          0.00000        0.99181   -0.97804
LowBirthWeight  0.99181        0.00000    0.99616
Population     -0.97804        0.99616    0.00000

Thus, after removing the covariation we conclude there is indeed a strong correlation between lead and low birth weights.

Partial correlations:
                LowBirthWeight Population
HiLead                 0.99181   -0.97804
LowBirthWeight                    0.99616

Another example

Do Democrats prefer cats? The question I was interested in, Do liberals really prefer cats, was inspired by a Time magazine 18 February 2014 article. I collated data on a separate but related question: Do states with more registered Democrats have more cat owners? The data set was compiled from three sources: 2010 USA Census, a 2011 Gallup poll about religious preferences, and from a data book on survey results of USA pet owners (data at end of this page, scroll or click here to access the data).

This data set also demonstrates use of transformations of the data to improve fit of the data to statistical assumptions (normality, homoscedacity).

The variables, and their definitions, were:

ASDEMS = DEMOCRATS. Democrat advantage: the difference in registered Democrats compared to registered Republicans as a percentage; to improve the distribution qualities the arcsine transform was applied..

ASRELIG = RELIGION. Percent Religious from a Gallup poll who reported that Religion was “Very Important” to them. Also arcsine-transformed to improve normality and homoescedasticity (there you go, throwing $3 words around 🤠 ).

LGCAT = Number of pet cats, log₁₀-transformed, estimated for USA states by survey, except Alaska and Hawaiʻi (not included in the survey by the American Veterinary Association).

LGDOG = Estimated number of pet dogs, log₁₀-transformed for states, except Alaska and Hawaiʻi (not included in the survey by the American Veterinary Association).

LGIPC = Per capita income, log₁₀-transformed.

LGPOP = Population size of each state, log₁₀ transformed.

As always, begin with data exploration. All of the variables were right-skewed, so I applied data transformation functions as appropriate: log₁₀ for the quantitative data and arc sine transform for the frequency variables. Because Democrat Advantage and Percent Religious variables were in percentages, the values were first divided by 100 to make frequencies, then the R function asin() was applied. All analyses were conducted on the transformed data, therefore conclusions apply to the transformed data.

A scatter plot matrix (KMggplo2) plus histograms of the variables along the diagonals shows the results of the transforms and hints at the associations among the variables. A graphic like this one is called a trellis plot; a layout of smaller plots in a grid with the same (preferred) or at least similar axes. Trellis plots (Fig 5) are useful for finding the structure and patterns in complex data. Scanning across a row shows relationships between one variable with all of the others. For example, the first row Y-axis is for the ASDEMS variable; from left to right along the row we have, after the histogram, what look to be weak associations between ASDEMS and ASRELIG, LGCAT, LGDOG, and LGDOG.

Figure 5. Trellis plot, correlations among variables.

A matrix of partial correlations was produced from the Rcmdr correlation call. Thus, to pick just one partial correlation, the association between DEMOCRATS and RELIGION (reported as “very important”) is negative (r = -0.45) and from the second matrix we retrieve the approximate P-value, unadjusted for the multiple comparisons problem, of p = 0.0024. We quickly move past this matrix to the adjusted P-values and confirm that this particular correlation is statistically significant even after correcting for multiple comparisons. Thus, there is a moderately strong negative correlation between those who reported that religion was very important to them and the difference between registered Democrats and Republicans in the 48 states. Because it is a partial correlation, we can conclude that this correlation is independent of all of the other included variables.

And what about our original question: Do Democrats prefer cats over dogs? The partial correlation after adjusting for all of the other correlated variables is small (r = 0.05) and not statistically different from zero (P-value greater than 5%).

Are there any interesting associations involving pet ownership in this data set? See if you can find it (hint: the correlation you are looking for is also in red).

Partial correlations:
         ASDEMS  ASRELIG LGCAT   LGDOG   LGIPC   LGPOP
ASDEMS   0.0000 -0.4460  0.0487  0.0605  0.1231 -0.0044
ASRELIG -0.4460  0.0000 -0.2291 -0.0132 -0.4685  0.2659
LGCAT    0.0487 -0.2291  0.0000  0.2225 -0.1451  0.6348
LGDOG    0.0605 -0.0132  0.2225  0.0000 -0.6299  0.5953
LGIPC    0.1231 -0.4685 -0.1451 -0.6299  0.0000  0.6270
LGPOP   -0.0044  0.2659  0.6348  0.5953  0.6270  0.0000

Raw P-values, Pairwise two-sided p-values:

         ASDEMS ASRELIG  LGCAT   LGDOG   LGIPC   LGPOP
ASDEMS           0.0024  0.7534  0.6965  0.4259  0.9772
ASRELIG  0.0024          0.1347  0.9325  0.0013  0.0810
LGCAT    0.7534  0.1347          0.1465  0.3473  <.0001
LGDOG    0.6965  0.9325  0.1465          <.0001  <.0001
LGIPC    0.4259  0.0013  0.3473  <.0001          <.0001
LGPOP    0.9772  0.0810  <.0001  <.0001  <.0001

Adjusted P-values, Holm’s method (Benjamini and Hochberg 1995)

         ASDEMS ASRELIG  LGCAT   LGDOG   LGIPC   LGPOP
ASDEMS           0.0241  1.0000  1.0000  1.0000  1.0000
ASRELIG  0.0241          1.0000  1.0000  0.0147  0.7293
LGCAT    1.0000  1.0000          1.0000  1.0000  <.0001
LGDOG    1.0000  1.0000  1.0000          <.0001  0.0002
LGIPC    1.0000  0.0147  1.0000  <.0001          <.0001
LGPOP    1.0000  0.7293  <.0001  0.0002  <.0001

A graph (Fig 6) to summarize the partial correlations: green lines indicate positive correlation, red lines show negative correlations; Strength of association indicated by the line thickness, with thicker lines corresponding to greater correlation.

Figure 6. Causal paths among variables.

As you can see, partial correlation analysis is good for a few variables, but as the numbers increase it is difficult to make heads or tails out of the analysis. Better methods for working with these highly correlated data in what we call multivariate data analysis, for example Structural Equation Modeling or Path Analysis.

Questions

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help.
2. Spurious correlations can be challenging to recognize, and, sometimes, they become part of a challenge to medicine to explain away. A classic spurious correlation is the correlation between rates of MMR vaccination and autism prevalence. Here’s a table of numbers for you

3. Make scatterplots of autism prevalence vs
   • Herb supplement revenue
   • Fertility rate
   • MMR vaccination
   • UFC revenue
4. Calculate and test correlations between autism prevalence vs
   • Herb supplement revenue
   • Fertility rate
   • MMR vaccination
   • UFC revenue
5. Interpret the correlations — is there any clear case for autism vs MMR?
6. What additional information is missing from Table 2? Add that missing variable and calculate partial correlations for autism prevalence vs
   • Herb supplement revenue
   • Fertility rate
   • MMR vaccination
   • UFC revenue
7. Do a little research: What are some reasons for increase in autism prevalence? What is the consensus view about MMR vaccine and risk of autism?

Quiz Chapter 16.2

Causation and Partial correlation

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Data sets

Data used in this page, 100 meter running times since 1900.

Data used in this page, birth weight by lead exposure

Data in this page, Do Democrats prefer cats?

Chapter 16 contents

• Introduction
• Product moment correlation
• Causation and Partial correlation
• Data aggregation and correlation
• Spearman and other correlations
• Instrument reliability and validity
• Similarity and Distance
• References and suggested readings

15.3 – Wilcoxon signed rank test

Introduction

When experimental units are repeated or paired, they lack independence and evaluating any difference between paired groups without accounting for the association between the repeat measures or between the pairs of related subjects would lead to incorrect inferences. A familiar pairing of experimental units occurs in clinical observational research in which control subjects and treatment subjects are matched by many characteristics.

In such cases, the parametric paired t-test would be used to evaluate inferences about the differences between repeat measures or between treatment and matched control subjects for some measured outcome. A nonparametric alternative to the paired t-test is the Wilcoxon signed rank test, also called paired Wilcoxon test.

Another common example of paired sampling units would be that individuals are measured more than once for the same character or feature. For example, Chapter 10.3 we presented results of running pace in minutes to complete the race of 15 women for repeated trials (different years) at a 5K race held annually in Honolulu (Table 1).

Table 1. 5K repeat measures running data. 

To get the paired Wilcoxon test in R Commander, select Rcmdr: Statistics → Nonparametric tests → Paired Wilcoxon Test 

Figure 1. R Commander paired Wilcoxon test menu (aka Wilcoxon signed rank sum test). Rcmdr version 2.7.

Select first variable (e.g., Race1), second variable (e.g., Race2). Next, select Options tab and set null hypothesis. Accept defaults (Fig. 2).

Figure 2. R Commander Options, select null hypothesis.

For the nonparametric paired Wilcoxon test we choose among the options to set our conditions; from the context menu a two-tailed test and we elect to use the defaults for the tests and calculations of P-values.

The results, copied from the Output window, are shown below. The calculation of the Wilcoxon test statistic (V) is straightforward, involving summing the ranks. Obtaining the P-value of the test of the null is a bit more involved as it depends on permutations of all possible combinations of differences. For us, R will do nicely with the details, and we just need to check the P-value.

with(repeat15_banana5K, wilcox.test(Race.1, Race.2, alternative='two.sided', paired=TRUE))
# median difference
[1] -0.3313126

Wilcoxon signed rank exact test
data: Race.1 and Race.2
V = 58, p-value = 0.9341
alternative hypothesis: true location shift is not equal to 0

End R output

Here, we see that the median difference is small (-0.33), and the associated P-value is 0.93. Thus, we failed to reject the null hypothesis and should conclude that there was no difference in median running pace during the first and second trials.

Note that this is the same general conclusion we got when we ran a paired t-test on this data set: no difference between day one and day two.

R code

example.ch10.3 <- read.table(header=TRUE, text = "
ID Race1 Race2
1 15.28 15.61
2 11.22 11.19
3 8.80 9.14
4 8.88 5.46
5 9.81 10.50
6 6.12 5.69
7 8.31 8.71
8 6.26 7.42
9 17.16 16.41
10 16.23 15.82
11 5.90 7.12
12 8.31 10.48
13 5.93 8.64
14 10.54 5.99
15 9.53 8.69
")

Questions

1. This question lists all fourteen statistical tests we have been introduced to so far
   a. Mark yes or no as to whether or not the test is a parametric test
   b. Identify the nonparametric test(s) with their equivalent parametric test(s). If there are no equivalency, simply write “none.”

Quiz Chapter 15.3

Wilcoxon Signed Rank Test

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Chapter 15 contents

• Introduction
• Kruskal-Wallis and ANOVA by ranks
• Wilcoxon Rank Sum Test
• Wilcoxon signed rank test
• References and suggested readings

15.1 – Kruskal-Wallis and ANOVA by ranks

Introduction

When the data are NOT normally distributed OR When the variances in the different samples are unequal, one option is to opt for a nonparametric alternative use the Kruskal-Wallis test. The test is the same as an ANOVA on ranks. The tests are used to test whether or not groups can be viewed as having come from the same sample distribution. This is a terse page with emphasis on examples.

It is known that the an ANOVA on ranks (in blue) of the original data (in black) will yield the same results.

Kruskal-Wallis

Rcmdr: Statistics → Nonparametric test → Kruskal-Wallis test…

Rcmdr Output of Kruskal-Wallis test

tapply(Pop_data$Stuff, Pop_data$Pop, median, na.rm=TRUE)

Pop1 Pop2 Pop3 Pop4 
 146   90  122  347 
kruskal.test(Stuff ~ Pop, data=Pop_data) 
Kruskal-Wallis rank sum test 
data: Stuff by Pop 
Kruskal-Wallis chi-squared = 25.6048, df = 3, p-value = 1.154e-05

End of R output

So, we reject — “provisionally, of course! — the null hypothesis, right?

Compare parametric test and alternative non-parametric test

Let’s compare the nonparametric test results to those from an analysis of the ranks (ANOVA of ranks).

To get the ranks in R Commander (example.15.1 data set is available at bottom of this page; scroll down or click here).

Rcmdr: Data → Manage variables in active data set → Compute new variable …

The command for ranks is…. wait for it …. rank(). In the popup menu box, name the new variable (Ranks) and in the Expression to compute box enter rank(Values).

Figure 1. Screenshot Rcmdr menu create new variable.

And the R code is simply

example.15.1$Ranks <- with(example.15.1, rank(Values))

That’s one option, rank across the entire data set. Another option would be to rank within groups.

R code

example.15.1$xRanks <- ave(Values, Population, FUN=rank)

Here are the results of ranking within groups (Table 1).

Table 1. Original and ranked data set.

Question 1. Which do you choose, rank across groups (Ranks) or rank within groups (xRanks)? Recall that this example began with a nonparametric alternative to the one-way ANOVA, and we were testing the null hypothesis that the group means were the same

Answer 1. Rank the entire data set, ignoring the groups. The null hypothesis here is that there is no difference in median ranks among the groups. Ranking within groups simply shuffles observations within the group. This is basically the same thing as running Kruskal-Wallis test.

Run the one-way ANOVA, now on the Ranked variable. The ANOVA table is summarized Table 2.

Table 2. ANOVA table on ranked variable.

So, how do you choose between the parametric ANOVA and the non-parametric Kruskal-Wallis (ANOVA by Ranks) test? Think like a statistician — It is all about the type I error rate and potential bias of a statistical test. The purpose of statistics is to help us separate real effects from random chance differences. If we are employing the NHST approach, then we must consider our chance that we are committing either a Type I error or a Type II error, and conservative tests, eg, tests based on comparing medians and not means, implies an increased chance of committing Type II errors.

Questions

1. Saying that nonparametric tests make fewer assumptions about the data should not be interpreted that they make no assumptions about the data. Thinking about our discussions about experimental design and our discussion about test assumptions, what assumptions must hold regardless of the statistical test used?
2. Go ahead and carry out the one-way ANOVA on the within group ranks (xRanks). What’s the p-value from the ANOVA?
3. One could take the position that only nonparametric alternative tests should be employed in place of parametric tests, in part because they make fewer assumptions about the data. Why is this position unwarranted?

Quiz Chapter 15.1

Kruskal-Wallis and ANOVA by ranks

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Data used in this page

Dataset for Kruskal-Wallis test

Chapter 15 contents

• Introduction
• Kruskal-Wallis and ANOVA by ranks
• Wilcoxon Rank Sum Test
• Wilcoxon signed rank test
• References and suggested readings

20.10 – Growth equations and dose response calculations

Introduction

In biology, growth may refer to increase in cell number, change in size of an individual across development, or increase of number of individuals in a population over time. Nonlinear, time-series, several models proposed to fit growth data, including the Gompertz, logistic, and the von Bertalanffy. These models fit many S-shaped growth curves. These models are special cases of generalized linear models, also called Richard curves.

Growth example

This page describes how to use R to analyze growth curve data sets.

Key to parameter estimates: y0 is the lag, mumax is the growth rate, and K is the asymptotic stationary growth phase. The spline function does not return an estimate for K.

R code

require(growthrates)
#Enter the data. Replace these example values with your own 
#time variable (Hours)
Hours <- c(0.000, 0.274, 0.384, 0.658, 0.986, 1.260, 1.479, 1.699, 1.973, 2.137, 2.466, 2.795, 3.123, 3.671, 4.110, 4.548, 5.151, 5.534, 5.863, 6.521, 7.068, 7.671, 8.164, 8.877, 9.644, 10.466, 11.233, 11.890, 12.493, 13.205, 14.082)
#absorbance or concentration variable (Abs)
Abs <- c(0.002207, 0.010443, 0.033688, 0.063257, 0.111848, 0.249240, 0.416236, 0.515578, 0.572632, 0.589528, 0.619091, 0.608486, 0.621136, 0.616850, 0.614689, 0.614643, 0.612465, 0.606082, 0.603933, 0.595407, 0.589006, 0.578372, 0.567749, 0.559217, 0.546451, 0.537907, 0.537826, 0.529300, 0.516551, 0.505905, 0.491013)

#Make a dataframe and check the data; If error, then check that variables have equal numbers of entries
Yeast <- data.frame(Hours,Abs); Yeast

#Obtain growth parameters from fit of a parametric growth model
#First, try some reasonable starting values
p <- c(y0 = 0.001, mumax = 0.5, K = 0.6)
model.par <- fit_growthmodel(FUN = grow_logistic, p = p, Hours, Abs, method=c("L-BFGS-B"))
summary(model.par)
coef(model.par)

#Obtain growth parameters from fit of a nonparametric smoothing spline
model.npar <- fit_spline(Hours,Abs)
summary(model.npar)
coef(spline.md)

#Make plots
par(mfrow = c(2, 1))
plot(Yeast, ylim=c(0,1), cex=1.5,pch=16, main="Parametric Nonlinear Growth Model", xlab="Hours", ylab="Abs")
lines(model.par, col="blue", lwd=2)
plot(model.npar, ylim=c(0,1), lwd=2, main="Nonparametric Spline Fit", xlab="Hours", ylab="Abs")

Results from example code

Parameters:
      Estimate Std. Error t value Pr(>|t|)    
y0    0.003765   0.001049    3.59  0.00125 ** 
mumax 3.954458   0.238810   16.56 5.37e-16 ***
K     0.579354   0.008092   71.60  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.03833 on 28 degrees of freedom

Parameter correlation:
            y0   mumax        K
y0     1.00000 -0.8755  0.05847
mumax -0.87550  1.0000 -0.14623
K      0.05847 -0.1462  1.00000
         y0       mumax           K 
0.003764518 3.954457995 0.579354378 
Fitted smoothing spline:
Call:
smooth.spline(x = time, y = ylog)

Smoothing Parameter  spar= 0.4409789  lambda= 1.221762e-05 (12 iterations)
Equivalent Degrees of Freedom (Df): 14.70929
Penalized Criterion (RSS): 0.5024203
GCV: 0.05868796

Parameter values of exponential growth curve:
Maximum growth at x= 0.0004255438 , y= 0.003072228 
y0 = 0.003065586 
mumax = 5.085845 

r2 of log transformed data= 0.9906254 
         y0       mumax 
0.003065586 5.085845036

Figure 1. Top: Parametric Nonlinear Growth Model; Bottom: Nonparametric Spline Fit

LD₅₀

In toxicology, the dose of a pathogen, radiation, or toxin required to kill half the members of a tested population of animals or cells is called the lethal dose, 50%, or LD₅₀. This measure is also known as the lethal concentration, LC₅₀, or properly after a specified test duration, the LCt₅₀ indicating the lethal concentration and time of exposure. LD₅₀ figures are frequently used as a general indicator of a substance’s acute toxicity. A lower LD₅₀ is indicative of increased toxicity.

The point at which 50% response of studied organisms to range of doses of a substance (e.g., agonist, antagonist, inhibitor, etc.) to any response, from change in behavior or life history characteristics up to and including death can be described by the methods described in this chapter. The procedures outlined below assume that there is but one inflection point, i.e., an “s-shaped” curve, either up or down; if there are more than one inflection points, then the logistic equations described will not fit the data well and other choices need to be made (see Di Veroli et al 2015). We will use the drc package (Ritz et al 2015).

Example

First we’ll work through use of R. We’ll follow up with how to use Solver in Microsoft Excel.

After starting R, load the drc library.

library(drc)

Consider some hypothetical 24-hour survival data for yeast exposed to salt solutions. Let resp equal the variable for frequency of survival (e.g., calculated from OD₆₆₀ readings) and NaCl equal the millimolar (mm) salt concentrations or doses.

At the R prompt type

resp <- c(1,1,1,.9,.7,.3,.4,.2,0,0,0)
NaCl=seq(0,1000,100) 
#Confirm sequence was correctly created; alternatively, enter the values.
NaCl 
[1] 0 100 200 300 400 500 600 700 800 900 1000 
#Make a plot
plot(NaCl,resp,pch=19,cex=1.2,col="blue",xlab="NaCl [mm]",ylab="Survival frequency")

And here is the plot (Fig. 2).

Figure 2. Hypothetical data set, survival of yeast in different salt concentrations.

Note the sigmoidal “S” shape — we’ll need an logistic equation to describe the relationship between survival of yeast and NaCl doses.

The equation for the four parameter logistic curve, also called the Hill-Slope model, is

where c is the parameter for the lower limit of the response, d is the parameter for the upper limit of the response, e is the relative EC₅₀, the dose fitted half-way between the limits c and d, and b is the relative slope around the EC₅₀. The slope, b, is also known as the Hill slope. Because this experiment included a dose of zero, a three parameter logistic curve would be appropriate. The equation simplifies to

EC₅₀ from 4 parameter model

Let’s first make a data frame

dose <- data.frame(NaCl, resp)

Then call up a function, drm, from the drc library and specify the model as the four parameter logistic equation, specified as LL.4(). We follow with a call to the summary command to retrieve output from the drm function. Note that the four-parameter logistic equation

model.dose1 = drm(dose,fct=LL.4())
summary(model.dose1)

And here is the R output.

Model fitted: Log-logistic (ED50 as parameter) (4 parms) 
Parameter estimates: 
                   Estimate  Std. Error    t-value  p-value 
b:(Intercept)      3.753415    1.074050   3.494636   0.0101 
c:(Intercept)     -0.084487    0.127962  -0.660251   0.5302 
d:(Intercept)      1.017592    0.052460  19.397441   0.0000 
e:(Intercept)    492.645128   47.679765  10.332373   0.0000 

Residual standard error: 
0.0845254 (7 degrees of freedom)

The EC₅₀, technically the LD₅₀ because the data were for survival, is the value of e: 492.65 mM NaCl.

You should always plot the predicted line from your model against the real data and inspect the fit.

At the R prompt type

plot(model.dose1, log="",pch=19,cex=1.2,col="blue",xlab="NaCl [mm]",ylab="Survival frequency")

As long as the plot you made in earlier steps is still available, R will add the line specified in the lines command. Here is the plot with the predicted logistic line displayed (Fig. 3).

Figure 3. Logistic curve added to Figure 1 plot.

While there are additional steps we can take to decide is the fit of the logistic curve was good to the data, visual inspection suggests that indeed the curve fits the data reasonably well.

More work to do

Because the EC₅₀ calculations are estimates, we should also obtain confidence intervals. The drc library provides a function called ED which will accomplish this. We can also ask what the survival was at 10% and 90% in addition to 50%, along with the confidence intervals for each.

At the R prompt type

ED(model.dose1,c(10,50,90), interval="delta")

And the output is shown below.

Estimated effective doses
(Delta method-based confidence interval(s))
     Estimate Std. Error   Lower   Upper
1:10  274.348     38.291 183.803  364.89
1:50  492.645     47.680 379.900  605.39
1:90  884.642    208.171 392.395 1376.89

Thus, the 95% confidence interval for the EC₅₀ calculated from the four parameter logistic curve was between the lower limit of 379.9 and an upper limit of 605.39 mm NaCl.

EC₅₀ from 3 parameter model

Looking at the summary output from the four parameter logistic function we see that the value for c was -0.085 and p-value was 0.53, which suggests that the lower limit was not statistically different from zero. We would expect this given that the experiment had included a control of zero mm added salt. Thus, we can explore by how much the EC₅₀ estimate changes when the additional parameter c is no longer estimated by calculating a three parameter model with LL.3().

model.dose2 = drm(dose,fct=LL.3())
summary(model.dose2)

R output follows.

Model fitted: Log-logistic (ED50 as parameter) with lower limit at 0 (3 parms)
Parameter estimates:
               Estimate Std. Error   t-value p-value
b:(Intercept)   4.46194    0.76880   5.80378   4e-04
d:(Intercept)   1.00982    0.04866  20.75272   0e+00
e:(Intercept) 467.87842   25.24633  18.53253   0e+00
Residual standard error:
 0.08267671 (8 degrees of freedom)

The EC₅₀ is the value of e: 467.88 mM NaCl.

How do the four and three parameter models compare? We can rephrase this as as statistical test of fit; which model fits the data better, a three parameter or a four parameter model?

At the R prompt type

anova(model.dose1, model.dose2)

The R output follows.

1st mode
fct:      LL.3()
2nd model
fct:      LL.4()
ANOVA table
          ModelDf      RSS Df F value p value
2nd model       8 0.054684                   
1st model       7 0.050012  1  0.6539  0.4453

Because the p-value is much greater than 5% we may conclude that the fit of the four parameter model was not significantly better than the fit of the three parameter model. Thus, based on our criteria we established in discussions of model fit in Chapter  16 – 18, we would conclude that the three parameter model is the preferred model.

The plot below now includes the fit of the four parameter model (red line) and the three parameter model (green line) to the data (Fig. 4).

Figure 4. Four parameter (red) and three parameter (green) logistic models fitted to data.

The R command to make this addition to our active plot was

lines(dose,predict(model.dose2, data.frame(x=dose)),col="green")

We continue with our analysis of the three parameter model and produce the confidence intervals for the EC₅₀ (modify the ED() statement above for model.dose2 in place of model.dose1).

Estimated effective doses
(Delta method-based confidence interval(s))
     Estimate Std. Error   Lower  Upper
1:10  285.937     33.154 209.483 362.39
1:50  467.878     25.246 409.660 526.10
1:90  765.589     63.026 620.251 910.93

Thus, the 95% confidence interval for the EC₅₀ calculated from the three parameter logistic curve was between the lower limit of 409.7 and an upper limit of 526.1 mm NaCl. The difference between upper and lower limits was 116.4 mm NaCl, a smaller difference than the interval calculated for the 95% confidence intervals from the four parameter model (225.5 mm NaCl). This demonstrates the estimation trade-off: more parameters to estimate reduces the confidence in any one parameter estimate.

Additional notes of EC₅₀ calculations

Care must be taken that the model fits the data well. What if we did not have observations throughout the range of the sigmoidal shape? We can explore this by taking a subset of the data.

dd = dose[1:6,]

Here, all values after dose 500 were dropped

dd
  resp dose
1  1.0    0
2  1.0  100
3  1.0  200
4  0.9  300
5  0.7  400
6  0.3  500

and the plot does not show an obvious sigmoidal shape (Fig. 5).

Figure 5. Plot of reduced data set.

We run the three parameter model again, this time on the subset of the data.

model.dosedd = drm(dd,fct=LL.3())
summary(model.dosedd)

Output from the results are

Model fitted: Log-logistic (ED50 as parameter) with lower limit at 0 (3 parms)
Parameter estimates:
                Estimate Std. Error    t-value p-value
b:(Intercept)   6.989842   0.760112   9.195801  0.0027
d:(Intercept)   0.993391   0.014793  67.153883  0.0000
e:(Intercept) 446.882542   5.905728  75.669344  0.0000
Residual standard error:
 0.02574154 (3 degrees of freedom)

Conclusion? The estimate is different, but only just so, 447 vs. 468 mm NaCl.

Thus, within reason, the drc function performs well for the calculation of EC₅₀. Not all tools available to the student will do as well.

NLopt and nloptr

draft

Free open source library for nonlinear optimization.

Steven G. Johnson, The NLopt nonlinear-optimization package, http://github.com/stevengj/nlopt

https://cran.r-project.org/web/packages/nloptr/vignettes/nloptr.html

Alternatives to R

What about online tools? There are several online sites that will allow students to perform these kinds of calculations. Students familiar with MatLab know that it can be used to solve nonlinear equation problems. An open source alternative to MatLab is called GNU Octave, which can be installed on a personal computer or run online at http://octave-online.net. Students also may be aware of other sites, e.g., mycurvefit.com and IC50.tk.

Both of these free services performed well on the full dataset (results not shown), but fared poorly on the reduced subset: mycurvefit returned a value of 747.64 and IC50.tk returned an EC₅₀ estimate of 103.6 (Michael Dohm, pers. obs.).

Simple inspection of the plotted values shows that these values are unreasonable.

EC₅₀ calculations with Microsoft Excel

Most versions of Microsoft Excel include an add-in called Solver, which will permit mathematical modeling. The add-in is not installed as part of the default installation, but can be installed via the Options tab in the File menu for a local installation or via Insert for the Microsoft Office online version of Excel (you need a free Microsoft account). The following instructions are for a local installation of Microsoft Office 365 and were modified from information provided by sharpstatistics.co.uk.

After opening Excel, set up worksheet as follows. Note that in order to use my formulas your spreadsheet values need to be set up exactly as I describe.

• Dose values in column A, beginning with row 2
• Response values in column B, beginning with row 2
• In column F type b in row 2, c in row 3, d in row 4, and e in row 5.
• In cells G2 – G5, enter the starting values. For this example, I used b = 1, c = 0, d = 1, e = 400.
  • For your own data you will have to explore use of different starting values.
• Enter headers in row 1.
  • Column A Dose
  • Column B Response
  • Column C Predicted
  • Column D Squared difference
  • Column F Constants
• In cell C14 enter sum squares
• In cell D14 enter =sum(D2:D12)

Here is an image of the worksheet (Fig. 6), with equations entered and values updated, but before Solver run.

Figure 6. Screenshot Microsoft Excel worksheet containing our data set (col A & B), with formulas added and calculated. Starting values for constants in column G, rows 2 – 4.

Next, enter the functions.

• In cell C2 enter the four parameter logistic formula — type everything between the double quotes: “=$G$3+(($G$4-$G$3)/(1+(A2/$G$5)^$G$2))“. Next, copy or drag the formula to cell C12 to complete the predictions.
  • Note: for a three parameter model, replace the above formula with “=(($G$4)/(1+(A2/$G$5)^$G$2))“.
• In cell D2 type everything between the double quotes: “=(B2-C2)^2“.
• Next, copy or drag the formula to cell D12 to complete the predictions.
• Now you are ready to run solver to estimate the values for b, c, d, and e.
  • Reminder: starting values must be in cells G2:G5
• Select cell D14 and start solver by clicking on Data and looking to the right in the ribbon.

Solver is not normally installed as part of the default installation of Microsoft Excel 365. If the add-in has been installed you will see Solver in the Analyze box (Fig. 7).

Figure 7. Screenshot Microsoft Excel, Solver add-in available.

If Solver is not installed, go to File → Options → Add-ins (Fig. 8).

Figure 8. Screenshot Microsoft Excel, Solver add-in available and ready for use.

Go to Microsoft support for assistance with add-ins.

With the spreadsheet completed and Solver available, click on Solver in the ribbon (Fig. 7) to begin. The screen shot from the first screen of Solver is shown below (Fig. 9).

Figure 9. Screenshot Microsoft Excel solver menu.

Setup Solver

• Set Objective, enter the absolute cell reference to the sum squares value, \$D$14
• Set To: Min.
• For By Changing Variable Cells, enter the range of cells for the four parameters, $G$2:$G$5
• Uncheck the box by “Make Unconstrained Variables Non-Negative.”
  • Where constraints refers to any system of equalities or inequalities equations imposed on the algorithm.
• Select a Solving method, choose “GRG Nonlinear,” the nonlinear programming solver option (not shown in Fig. 7, select by clicking on the down arrow).
• Click Solve button to proceed.

If all goes well, this next screen (Fig. 10) will appear which shows the message, “Solver has converged to the current solution. All constraints are satisfied.”

Figure 10. Screenshot solver completed run.

Click OK and note that the values of the parameters have been updated (Table 1).

Table 1. Four parameter logistic model, results from Solver 

Note the values obtained by Solver are virtually identical to the values obtained in the drc R package. The differences are probably because of the solver algorithm.

More interestingly, how did Solver do on the subset data set? Here are the results from a three parameter logistic model (Table 2).

Table 2. Three parameter logistic model, results from Solver

The results are again very close to results from the drc R package.

Thus, we would conclude that Solver and Microsoft Excel would be a reasonable choice for EC₅₀ calculations, and much better than IC50.tk and mycurvefit.com. The advantages of R over Microsoft Excel is that the model building is more straightforward than entering formulas in the cell reference format required by Excel.

Questions

[pending]

References and suggested readings

Beck B, Chen YF, Dere W, et al. Assay Operations for SAR Support. 2012 May 1 [Updated 2012 Oct 1]. In: Sittampalam GS, Coussens NP, Nelson H, et al., editors. Assay Guidance Manual [Internet]. Bethesda (MD): Eli Lilly & Company and the National Center for Advancing Translational Sciences; 2004-. Available from: www.ncbi.nlm.nih.gov/books/NBK91994/

Di Veroli G. Y., Fornari C., Goldlust I., Mills G., Koh S. B., Bramhall J. L., Richards, F. M., Jodrell D. I. (2015) An automated fitting procedure and software for dose-response curves with multiphasic features. Scientific Reports 5: 14701.

Ritz, C., Baty, F., Streibig, J. C., Gerhard, D. (2015) Dose-Response Analysis Using R PLOS ONE, 10(12), e0146021

Tjørve, K. M., & Tjørve, E. (2017). The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family. PloS one, 12(6), e0178691.

Chapter 20 contents

• Additional topics
• Area under the curve
• Peak detection
• Baseline correction
• Surveys
• Time series
• Dimensional analysis
• Estimating population size
• Diversity indexes
• Survival analysis
• Growth equations and dose response calculations
• Plot a Newick tree
• Phylogenetically independent contrasts
• How to get the distances from a distance tree
• Binary classification
• Meta-analysis

/MD

20.9 – Survival analysis

draft

Introduction

As the name suggests, survival analysis is a branch of statistics used to account for death of organisms, or more generally, failure in a system.  In general, this kind of analysis models time to event, where event would be death or failure.  The basics of the method is defined by the survival function

where t is time, T is a variable that represents time of death or other end point, and Pr is probability of an event occurring later than at time t.

Excellent resources available, series of articles in volume 89 of British Journal of Cancer: Clark et al (2003a), Bradburn et al (2003a), Bradburn et al (2003b), and Clark et al (2003b).

Hazard function

Defined as the event rate at time t based on survival for time times equal to or greater than t.

Censoring

Censoring is a a missing data problem typical of survival analysis.  Distinguish right-censored and left-censored.

Kaplan-Meier plot

Kaplan-Meier (KM) estimator of survival function. Other survival function estimators Fleming-Harrington. The KM estimator, is

where d_i is the number of events that occurred at time t_i, n_i is the number of individuals known to have survived or not been censored. Because it’s an estimate, a statistic, need an estimate of the error variance. Several options, the default in R is the Greenwood estimator.

The KM plot, censoring times noted with plus.

R code

Download and install the RcmdrPlugin.survival package.

Example

data(heart, package="survival")
attach(heart)
#Get help with the data set
help("heart", package="survival")

head(heart)

 start stop event        age      year surgery transplant id
1    0   50     1 -17.155373 0.1232033       0          0  1
2    0    6     1   3.835729 0.2546201       0          0  2
3    0    1     0   6.297057 0.2655715       0          0  3
4    1   16     1   6.297057 0.2655715       0          1  3
5    0   36     0  -7.737166 0.4900753       0          0  4
6   36   39     1  -7.737166 0.4900753       0          1  4

Run basic survival analysis. After installing the RcmdrPlugin.survival, from Rcmdr select estimate survival function.

Figure 1. Screenshot of menu call for survival analysis in Rcmdr

Get survival estimator and KM plot (Figure 2)

R output

.Survfit <- survfit(Surv(start, event) ~ 1, conf.type="log", conf.int=0.95, 
Rcmdr+ type="kaplan-meier", error="greenwood", data=heart)

 .Survfit
Call: survfit(formula = Surv(start, event) ~ 1, data = heart, error = "greenwood", 
conf.type = "log", conf.int = 0.95, type = "kaplan-meier")

n events median 0.95LCL 0.95UCL 
172 75 26 17 37

plot(.Survfit, mark.time=TRUE) 

quantile(.Survfit, quantiles=c(.25,.5,.75))
$quantile
25 50 75 
3 26 67

$lower
25 50 75 
1 17 46

$upper
25 50 75 
12 37 NA

#by default, Rcmdr removes the object
remove(.Survfit)

Figure 2. Kaplan-Meier plot of heart data. Dashed lines are upper and lower confidence intervals about the survival function.

ylim = c(0,1), ylab="Survival probability", xlab="Days", lwd=2, col="blue"

the data set includes age and whether or not subjects had heart surgery before transplant. Compare.

Variable surgery is recorded 0,1, so need to create a factor

fSurgery <- as.factor(surgery)

Now, to compare

Rcmdr: Statistics → Survival analysis → Compare survival functions…

R output

Rcmdr> survdiff(Surv(start,event) ~ fSurgery, rho=0, data=heart)
Call:
survdiff(formula = Surv(start, event) ~ fSurgery, data = heart, 
rho = 0)

              N Observed Expected (O-E)^2/E (O-E)^2/V
fSurgery=No 143       66     58.7     0.902      4.56
fSurgery=Yes 29        9     16.3     3.255      4.56

Chisq= 4.6 on 1 degrees of freedom, p= 0.03

Get the KM estimator and make a KM plot

mySurvfit <- survfit(Surv(start, event) ~ surgery, conf.type="log", conf.int=0.95,
type="kaplan-meier", error="greenwood", data=heart)

plot(mySurvfit, mark.time=TRUE, ylim=c(0,1),lwd=2, col=c("blue","red"), xlab="Number of days", ylab="Survival probability")
legend("topright", legend = paste(c("Surgery - No", "Surgery - Yes")), col = c("blue", "red"), pch = 19, bty = "n")

Figure 3. KM plot

The comparison plot can be made in Rcmdr by selecting our Surgery factor in Strata setting (Fig. 4). Recall that strata refers to subgroups of a population.

Figure 4. Screenshot of Survival estimator menu in Rcmdr.

Questions

[pending]

Chapter 20 contents

• Additional topics
• Area under the curve
• Peak detection
• Baseline correction
• Surveys
• Time series
• Dimensional analysis
• Estimating population size
• Diversity indexes
• Survival analysis
• Growth equations and dose response calculations
• Plot a Newick tree
• Phylogenetically independent contrasts
• How to get the distances from a distance tree
• Binary classification
• Meta-analysis

/MD

19 – Distribution free statistical methods

Introduction

We introduced the concept of permutation tests in our chapter on parameter estimates and statistical error (Chapter 3.3). Jackknife and bootstrapping are permutation approaches to working with data when the Central Limit theorem is unlikely to apply or, rather, we don’t wish to make that assumption. The jackknife is a sampling method involving repeatedly sampling from the original data set, but each time leaving one value out. The estimator, for example, the sample mean, is calculated for each sample. The repeated estimates from the jackknife approach yield many estimates which, collected, are used to calculate the sample variance. Jackknife estimators tend to be less biased than those from classical asymptotic statistics. Bootstrapping, and not jackknife resampling, is now the preferred permutation approach (add citations).

Bootstrapping

Bootstrapping involves large numbers of permutations of the data, which, in short, means we repeatedly take many samples of our data and recalculate our statistics on these sets of sampled data. We obtain statistical significance by comparing our result from the original data against how often results from our permutations on the resampled data sets exceed the originally observed results. By permutation methods, the goal is to avoid the assumptions made by large-sample statistical inference. Since its introduction, “bootstrapping” has been shown to be superior in many cases for statistics of error compared to the standard, classical approach (add citations).

Permutation vs classical NHST approach

There are many advocates for this approach, and, because we have computers now instead of the hand calculators our statistics ancestors used, permutation methods may be the approach you will take in your own work. However, the classical approach has it’s strengths; when the conditions, that is, when the assumptions of asymptotic statistics are met by the data, then the classical approaches tend to be less conservative than the permutation methods. By conservative, statisticians mean that a test performs at the level we expect it to. Thus, if the assumptions of classical statistics are met they return the correct answer more often than do the permutation tests.

Questions

[pending]

Chapter 19 contents

• Introduction
• 19.1 – Jackknife sampling
• 19.2 – Bootstrap sampling
• 19.3 – Monte Carlo methods
• Ch19 References and suggested readings

18 – Multiple Linear Regression

Introduction

This is the second part of our discussion about general linear models. In this chapter we extend linear regression from one (Chapter 17) to many predictor variables. We also introduce logistic regression, which uses logistic function to model binary outcome variables. Extensions to address ordinal outcome variables are also presented. We conclude with a discussion of model selection, which applies to models with two or more predictor variables.

For linear regression models with multiple predictors, in addition to our LINE assumptions, we add the assumption of no multicollinearity. That is, we assume our predictor variables are not themselves correlated.

Rcmdr and R have multiple ways to analyze linear regression models; we will continue to emphasize the general linear model approach, which allow us to handle continuous and categorical predictor variables.

Practical aspects of model diagnostics were presented in Chapter 17; these rules apply for multiple predictor variable models. Regression and correlation (Chapter 16) both test linear hypotheses: we state that the relationship between two variables is linear (the alternative hypothesis) or it is not (the null hypothesis). The difference? Correlation is a test of association (are variables correlated, we ask?), but are not tests of causation: we do not imply that one variable causes another to vary, even if the correlation between the two variables is large and positive, for example. Correlations are used in statistics on data sets not collected from explicit experimental designs incorporated to test specific hypotheses of cause and effect. Regression is to cause and effect as correlation is to association. Regression, ANOVA, and other general linear models are designed to permit the statistician to control for the effects of confounding variables provided the causal variables themselves are uncorrelated.

Models

Chapter 17 covered the simple linear model

Chapter 18 covers multiple regression linear model

where α or β₀ represent the Y-intercept and β or β₁, β₂, … β_n represent the regression slopes.

And the logistic regression model

where L refers to the upper or maximum value of the curve, refers to the rate of change at the steepest part of the curve, and refers to the inflection point of the curve. Logistic functions are S-shaped, and typical use involves looking at population growth rates (eg, Fig 1), or in the case of logistic regression, how a treatment effects the rate of growth.

Figure 1. Growth of bacteria over time (optical density at 600 nm UV spectrophotometer), fit by logistic function (dashed line).

Suggested homework

Homework 12: Multiple linear regression.

Quizzes in this chapter

A total of 59 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 18 contents

• Introduction
• Multiple linear regression
• Nonlinear regression
• Logistic regression
• Generalized Linear Squares
• Selecting the best model
• Compare two linear models
• References and suggested readings (Ch17 & 18)

17 – Linear Regression

Introduction

Regression is a toolkit for developing models of cause and effect between one ratio scale data type dependent response variables, and one (simple linear regression) or more or more (multiple linear regression) ratio scale data type independent predictor variables. By convention the dependent variable(s) is denoted by Y, the independent variable(s) represented by X₁, X₂, X_n for n independent variables. Like ANOVA, linear regression is simply a special case of the general linear model, first introduced in Chapter 12.7.

Components of a statistical model

Regression statistical methods return model estimates of the intercept and slope coefficients, plus statistics of regression fit (eg, R², aka “R-squared,” the coefficient of determination).

Chapter 17.1 – 17.9 cover the simple linear model

Chapter 18.1 – 18.5 cover the multiple regression linear model

where α or β₀ represent the Y-intercept and β or β₁, β₂, … β_n represent the regression slopes.

Regression and correlation test linear hypotheses

We state that the relationship between two variables is linear (the alternative hypothesis) or it is not (the null hypothesis). The difference? Correlation is a test of linear association (are variables correlated, we ask?), imply possible causation, but are not sufficient evidence for causation: we do not imply that one variable causes another to vary, even if the correlation between the two variables is large and positive, for example. Correlations are used in statistics on data sets not collected from explicit experimental designs incorporated to test specific hypotheses of cause and effect.

Linear regression, however, is to cause and effect as correlation is to association. With regression and ANOVA, we are indeed making a case for a particular understanding of the cause of variation in a response variable: modeling cause and effect is the goal. Regression, ANOVA, and other general linear models are designed to permit the statistician to control for the effects of confounding variables provided the causal variables themselves are uncorrelated.

When to use correlation and when to apply linear regression to data set? For two ratio scale variables, one can always apply either approach. Correlation is always appropriate if both variables are measured in the course of the study whereas a regression modeling approach would be implied where one variable was the result of manipulation by the researcher.

Assumptions of linear regression

The key assumption in linear regression is that a straight line indeed is the best fit of the relationship between dependent and independent variables. The additional assumptions of parametric tests (Chapter 13) also hold. In Chapter 18 we conclude with an extension of regression from one to many predictor variables and the special and important topic of correlated predictor variables or multicollinearity.

Build a statistical model, make predictions

In our exploration of linear regression we begin with simple linear regression, also called ordinary least squares regression, starting with one predictor variable. Practical aspects of model diagnostics are presented. Regression may be used to describe or to provide a predictive statistical framework. In Chapter 18 we conclude with an extension of regression from one to many predictor variables. We conclude with a discussion of model selection. Throughout, use of Rcmdr and R have multiple ways to analyze linear regression models are presented; we will continue to emphasize the general linear model approach, but note that use of linear model in Rcmdr provides a number of default features that are conveniently available.

Questions

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help.

Suggested homework

Homework 11: Simple linear regression.

Quiz

In addition to the quiz on this page, a total of 81 questions among the several subchapters, a mix of true or false and multiple choice question format.

Quiz Chapter 17

Linear Regression

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References

Linear regression is a huge topic; references I include are among my favorite on the subject, but are only a small and incomplete sampling. For simplicity, I merged references for Chapter 17 and Chapter 18 into one page at References and suggested readings (Ch17 & 18)

Chapter 17 contains

• Introduction
• Simple Linear Regression
• Relationship between the slope and the correlation
• Estimation of linear regression coefficients
• OLS, RMA, and smoothing functions
• Testing regression coefficients
• ANCOVA – analysis of covariance
• Regression model fit
• Assumptions and model diagnostics for Simple Linear Regression
• References and suggested readings (Ch17 & 18)

15 – Nonparametric tests

Introduction

T-tests and ANOVA are members of a statistical family of tests called parametric tests. Parametric tests assume that the

• sample of observations come from a particular probability distribution, eg, normal distribution.
• samples among groups have equal variances.

Assumptions for parametric tests were introduced in Chapter 13. In the case of the t-test and ANOVA, we assume that the samples come from a normal probability distribution and that the probabilities of the test statistic follow the t distribution or the F distribution, respectively.

Providing these assumptions hold, we can then proceed to interpret our results as if we are talking about the population as a whole from which the samples were selected.

In other words, the t-test asks (infers) about properties of a population, hence, we are asking about parameters of the population.

T-tests, ANOVA, and other parametric tests are designed to work with quantitative ratio-scale data types. If the data are of this type and the probability distribution is known, they are the best tests to use… they allow you to make conclusions about experiments at a defined Type I error rate of 5%.

But what if you can’t assume that distribution? Your options include

• transforming the data so as the data better meet the assumptions of parametric tests.
• apply nonparametric statistical tests.
• use of distribution-free resampling methods like the bootstrap as an alternative. Bootstrapping offers methods to estimate confidence intervals and perform hypothesis tests without relying on theoretical distributions.

Here, we introduce some of the classic nonparametric statistics as an alternative to parametric tests. Bootstrapping is introduced in Chapter 19.

Nonparametric tests make fewer assumptions

Nonparametric tests do not make the assumption about a particular distribution — distribution free tests — nor are they used to make inferences about population parameters. Instead, nonparametric tests are used when the data type are ranks (ordinal). Now, when you think about it, all quantitative data can be converted to ranks. Hence, this is the argument for why there are nonparametric alternatives for tests like the t-test. There are a number of nonparametric alternatives to parametric tests and we’ll list several in this chapter (randomization methods like the bootstrap, see Ch19.2, are also considered nonparametric — unless, of course the data is assumed to come from a particular distribution, then it’s called parametric bootstrap). Another nonparametric option is to run a permutation test on the data.

However, nonparametric tests lack statistical power

One downside for nonparametric tests is that they tend to have less statistical power compared to the parametric alternatives (see Chapter 11 for review of Statistical Power). Thus, nonparametric tests tend to have higher Type II rates of error — they fail to properly reject the null hypothesis when they should. This problem tends to be less important for large sample sizes.

Thus, this chapter covers some of the more popular nonparametric alternative tests. Chapter 19 – Distribution free statistical methods, highlights use of permutation and randomization approaches, which also are alternatives to parametric tests.

Suggested homework

Homework 9: Common nonparametric tests.

Quizzes in this chapter

A total of 18 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 15 contents

• Introduction
• Kruskal-Wallis and ANOVA by ranks
• Wilcoxon Rank Sum Test
• Wilcoxon signed rank test
• References and suggested readings

14.8 – More on the linear model in Rcmdr

Introduction

During the last lectures, we could have used the Two-Way ANOVA command in R or Rcmdr

R code: How to analyze multifactorial ANOVA problems

Rcmdr: Statistics → Means → Multiway ANOVA

To analyze our two factor data sets. As long as the design meets the following conditions, by all means use this command because it is simple and precisely correct.

• Both factors are fixed, not random.
• Each level of first factor is crossed with each level of the second factor.
• No missing data (the design is fully replicated and balanced).

If any of the three points do not fit your two-way design, then you’ll need a different, more general and powerful ANOVA procedure in R and Rcmdr to analyze these types of designs. You’ll need the lm() function (Fig. 1).

Rcmdr: Statistics → Fit models → Linear model…

Figure 1. Linear model menu in Rcmdr, version 2.7.0

Some R basics with the lm() function, the general linear model.

Response is specified by linear predictor(s), either factors or covariates (ratio-scale predictor variables). We communicate to R what the model is by using operators. The four more commonly used operators are

the basic way to include the model terms, i.e., the mode predictors

which is interpreted as the interaction of all the variables and the factors in the term

is interpreted as factor crossing

Not shown, but indicates the term on the left is nested within the term on the right.

A few examples: we’ll have three factors, A, B, and C. For our one-way ANOVA, the model specification would be

For our crossed, balanced two-way ANOVA, the model specification would be

or equivalently

And for our block ANOVA problem?

Click here to get entire List of model commands in R.

How would we analyze our snake experiment?

Table 1. The snake data set

We have two factors, but one factor is a Block (repeated measures on individuals). We need to tell R and Rcmdr what our model is. We’ll return to talk about models next time, a very important topic!! For now, think of a model as adding the independent variables together to predict the response variable.

In our Snake example, it’s a two-way ANOVA, but one factor is individual snake, the other is a treatment, and we have repeat measures, so there cannot be an interaction.

We tell R and Rcmdr which columns contain the Response, and under Model, we enter the columns with the two factors.

Figure 2. Menu of linear model with repeat measures model, Rcmdr, version 2.7.0.

You must also tell R and Rcmdr which factors in the model (if any) are random to get the correct F statistics. Almost without exception, blocking factors are always treated as Random

The output looks like this (see below). More complicated, true (which means more information!), but things marked in red we’ve seen before.

LinearModel.2 <- lm(flick ~ Snake +Source, data=L16SnakeTaste)

summary(LinearModel.2)

Call:
lm(formula = flick ~ Snake + Source, data = L16SnakeTaste)

Residuals:
Min 1Q Median 3Q Max 
-5.2222 -0.7222 0.0278 1.5694 7.4444

Coefficients:
             Estimate Std. Error t value Pr(>|t|) 
(Intercept)    -4.444      2.523  -1.762 0.10862 
Snake[T.2]      9.667      3.090   3.128 0.01072 * 
Snake[T.3]      3.000      3.090   0.971 0.35451 
Snake[T.4]     12.667      3.090   4.099 0.00215 ** 
Snake[T.5]      6.000      3.090   1.942 0.08085 . 
Snake[T.6]      6.333      3.090   2.050 0.06755 . 
Source[T.fish] 14.167      2.185   6.484 0.0000704 ***
Source[T.worm] 14.167      2.185   6.484 0.0000704 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.784 on 10 degrees of freedom
Multiple R-squared: 0.8858, Adjusted R-squared: 0.8058 
F-statistic: 11.08 on 7 and 10 DF, p-value: 0.0005337

Contrast coding

When a factor (categorical variable) is used in a linear model (lm() or aov()), R must convert the factor into one or more numeric predictor variables. Contrast coding tells us how to represent the factor in the model, how to interpret the coefficients associated with the factor, how the hypothesis tests of the coefficients are handled, and whether or not Type III sums of squares tests are valid. R can handle contrast coding in a couple of ways, either as a dummy contrast (each group if the factor is contrasted to a baseline) or effects coding, where group means are compared to the grand mean. The command summary(lm()) reports contrast coding that was used in the linear model. Every kind of contrast leads to the same fitted values and same overall F-test, but coefficient estimates and t-tests change.

For the ANOVA table, we call up the command via

Rcmdr: Models → Hypothesis tests → ANOVA table… (Fig. 3).

Figure 3. Rcmdr: Models → Hypothesis tests → ANOVA table… Rcmdr, version 2.7.0

Confirm that the model object is active (in this case, the object was LinearModel.2), accept the defaults about types of tests and marginality, and submit OK. The output is

Anova(LinearModel.2, type="II")
Anova Table (Type II tests)

Response: flick
          Sum Sq Df F value Pr(>F) 
Snake     307.61  5  4.2956 0.02396 * 
Source    802.78  2 28.0256 0.00007954 ***
Residuals 143.22 10 
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

End R output

You do not need to know all of the output; all of that output is there for a reason, of course, but for now, here’s what R and Rcmdr has to say (from the help menu):

The sequential sums of squares is the added sums of squares given that prior terms are in the model. These values depend upon the model order. The adjusted sums of squares are the sums of squares given that all other terms are in the model. These values do not depend upon the model order.

You should try our snake example again, but this time, remove the tongue flick response to the dH2O for the first snake — a missing value. (just type into the cell NA).

How would you use GLM to analyze a simple two-way ANOVA, a fully crossed, fully replicated (“balanced”) design? We could use the Two-way ANOVA command in R and Rcmdr, or we could use lm(). Try it with the data set from random vs nonrandom lecture.

Another example

Below, you see how the model is entered. To let R know that I wish R and Rcmdr to test the interaction, I need to add a Model term for that source of variation. I accomplish this by typing “Diet*Drug” (without the quotes).

Figure 4. Crossed, balanced design. Linear model menu, Rcmdr, version 1.9.2

After clicking OK, the following output from the lm function is returned. How does this compare to output from the two-way ANOVA command in R and Rcmdr?

You should try both and compare!

Rcmdr: Models → Hypothesis tests → ANOVA table

Anova(LinearModel.17, type="II")
Anova Table (Type II tests)
Response: chol_randomized
             Sum Sq  Df F value   Pr(>F) 
diet         141.08   2  1.3061 0.28745 
drug         351.88   2  3.2577 0.05403 .
diet:drug    235.50   4  1.0901 0.38124 
Residuals   1458.19  27

End R output

Nested ANOVA

The nested ANOVA may be analyzed in multiple ways in R and Rcmdr, but I prefer the lm() function because it is the most general. For Nested ANOVA, we can also use lm(). Here’s where it gets a little tricky. Put in the Response variable (Chol), then click in the box for model: Select both factors, then type in / after the factor that’s nesting factor. For our nested model example (14.5 – Nested designs), Manufacturer Source was nested within Drug.

Table 3. Nested design example data set from Chapter 14 – Nested design

Figure 5. Nested design, linear model menu, Rcmdr, version 1.9.2

Fortunately, R and Rcmdr’s help system is quite extensive here, so when in doubt, check the help box…

Output from the linear model for the Nested Example looks like the one below.

The General Linear Model function in R and therefore Rcmdr returns information about our design plus Sums of Squares, Mean squares, and P-values. R and Rcmdr default’s to use of sequential evaluation of effects. Adjusted evaluation is useful for when you have a covariate (like body size or another confounding variable) that should be evaluated first before the factors are evaluated. We will use the sequential analysis.

Rcmdr: Models → Hypothesis tests → ANOVA table

Anova(LinearModel.15, type="II")
Anova Table (Type II tests)
Response: Chol
             Sum Sq Df F value Pr(>F)
Drug         225.14  2  1.1743 0.3262
Drug:Source  269.27  3  0.9363 0.4385
Residuals   2300.61 24

End R output

Repeatability and ANOVA

We need to tell Rcmdr how to structure the error term; you need the data frame to be arranged so

aovRes <- aov(dH2O ~ Source + Error(Source/Subject), data=SnakeTaste)
#Print the results
aovRes
Anova Table (Type II tests)
Response: dH2O
            Sum Sq Df F value   Pr(>F)    
Subject     307.61  5  4.2956 0.02396 *  
Source      802.78  2 28.0256 0.00007954 ***
Residuals   143.22 10                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#What components are available in the aovRes object?
names(aovRes)
[1] "Sum Sq"  "Df"      "F value" "Pr(>F)"

How do I extract the “F value” for Subjects?

str(aovRes)

Questions

1. What are three learning objectives from this page?
2. Be able to distinguish use of lm() and aov() and the summary output with respect to testing of linear models.

Quiz Chapter 14.8

More on the linear model in Rcmdr

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Data sets

Table 1. The snake data set

Chapter 14 contents

• Introduction
• Crossed, balanced, fully replicated
• Sources of variation
• Fixed effects, Random effects
• Randomized block design
• Nested designs
• Some other ANOVA designs
• Rcmdr Multiway ANOVA
• More on the linear model in Rcmdr
• References and suggested readings