vital statistics

Final Marks

2014-01-01T16:42:00.001-05:00

Anna's Hummingbird PHOTO by Bruce Lyon

A few people have written to me about final marks. Relax. They will be posted on Solus sometime next week. Our secretaries have been away for the holidays and the deadline for submission to the registrar is not until tomorrow.

I hope you all had a good holiday and I am looking forward to seeing you either in courses, or just around campus in the coming year.

I have spent much of my time studying the hummingbird shown to the right. Not just that species but that very bird, who is just now sitting in a tree outside my window, while he waits for me to fill up the feeder. These birds are abundant here in Santa Cruz during the wintertime.

All the best for the New Year.

Project B due at noon tomorrow

2013-12-05T22:52:00.000-05:00

As I hope you are well aware, Project B is due before noon tomorrow, Friday 6 Dec. When marking this project we will give approximately equal weight to the following five elements of your report.

Abstract, Intro, Methods
Results (statistical analyses and reporting)
Discussion (some thoughtful discussion of what you found, what it means and any potential sources of error)
Figures and Tables (captions, axis labels, units, clarity, informative)
Style and dataset (organization, clarity, conciseness, presentation of material, quality of writing, avoidance of extraneous material, overall length not more than needed—there is no length requirement but your paper should not include too much extraneous material that is not needed)

Results and Discussion can be combined into one section, of course. I trust that there are no surprises here as these are all the elements of any good report.

Have a good holiday. I hope to see you again in the new year.

Bob Montgomerie

It's a Concept

2013-11-28T21:21:00.000-05:00

In class on Wednesday we talked about the important concepts that you should have learned and learned about in BIOL-243. If you take nothing more from this course, a clear understanding of these concepts will be very useful in the rest of your undergraduate career, and life!

They are, in no particular order except that I feel the the first one is most important:

Plot your data before you do any analysis
Think about your plotted data and what it shows
statistical significance
statistical vs biological significance
random sampling
sample vs population
normality
pseudoreplication
reporting the results of statistical analyses
hypothesis testing
sampling methods
independence
different kinds of test
parametric vs nonparametric analyses

Project Thoughts

2013-11-28T21:13:00.001-05:00

Several students have asked me more or less the same set of questions about Project B, so I thought it might be useful if I just address all of those questions here:

while there is no specific page length requirement. it seems to me that it should be easy to fit this project into 4 single-spaced pages, including graphs

single spaced
no need to make the graphs very big (7 cm x 5 cm is big enough and smaller could be good too
do not just screen capture ALL of the output from JMP, R or whatever stats program you are using; include only the info that is relevant to answering the questions and telling your story. If you don't refer to something in the text, then leave it out unless I specifically asked for it.

please include your dataset as a separate .csv file
the methods may not need to be longer than 3 sentences: 1. where you got the data, 2. how you sampled the data (or not), 3. what program you used to analyze the data The reader should be able to duplicate your study. No need to provide picky details about how you did analyses
Figures and tables need captions
we will be looking particularly for a clear demonstration that you understand the various concepts we outlined in class on Wednesday (see next post) and can express yourself and your analyses clearly and concisely
references are NOT required but I could see include a couple to support your conclusions, or point the reader to related studies
use whatever reference style you like. If in doubt pick a style used in any scientific journal
number the pages
do not over-report the results of your analyses, it is usually only necessary to say what test you used, the test statistic, the p-value, and the sample size
do not report more than 2 decimal places unless absolutely necessary (as in 0.00004)

Checking residuals

2013-11-28T21:02:00.000-05:00

When doing a regression analysis it's useful to check the residuals after you have done the regression. Two plots are useful, to help you verify that assumptions have been met.

First, check the plot of residuals against the predictor (X), like this one for the regression of Horsepower (Y) on Displacement (X) that I showed in class:

You want to check this graph to make sure that the distribution of residuals around the dotted line (zero) is not increasing or decreasing as you go from left to right. In the graph above the residuals look to be increasing but not too bad. You also want to make sure that they are uniformly scattered about the zero line and not curvilinear etc.

Nest it's useful to do a quantile-quantile plot of the residuals to test for normality. This is NOT the Residual Normal Quantile Plot that you get in JMP and is best plotted by saving the residuals (under the red triangle beside Line of Fit in JMP) as a new coloumn in the dataset, and then choosing Analyze>Distribution in JMP to get this:

As you can see the residuals are close to normal with maybe one outlier at the top of the graph.

Final Quiz

2013-11-25T17:00:00.001-05:00

The final quiz is available on Moodle and is due before 5 pm Tuesday. One student has reported not being able to see the whole set of graphs in question 1, so I have now made them smaller. DO not hesitate to contact me if there is anything like that that is preventing you from seeing the whole quiz. I do check the quizzes on my computer but my screen is probably bigger than yours.

Infographics

2013-11-24T13:52:00.001-05:00

Here is a selection of the infographics that I really liked from Project A. Several of them used the online info graphics maker Piktochart but not everyone who used Piktochart made a great info graphic. There were a few others that I liked so don't be too disappointed if yours is not here. These ones all seemed to me to be creative, informative and attractive. I have not identified the authors of these infographics as I think some of these people might prefer to be anonymous.

Prediction and prediction limits

2013-11-23T15:34:00.000-05:00

This information will give you a head start on Assignment 11 and part of Project B, for those of you trying to get finished early.

When you run a linear regression analysis you get a linear equation in the form Y = a + bX, where Y is the response, X is the predictor, a is the intercept (Y when X = 0) and b is the slope of the line. Here (to the right) is the result of a regression analysis I ran (in JMP) on some data on paternity in chinook salmon, collected by my collaborator and postdoc Patrice Rosengrave. In this study y = proportion of clutch sired by a male when he is competing with another male, and x is the proportion of the clutch he sires when there is no competitor. For convenience I will ref to these mostly as x and y here.

The equation for the linear regression line (in red) is y = 0.32 + 0.51x, and both the intercept and the slope are significantly different from zero, as you can see, although we are usually only interested ion testing the slope. In this sample the median of x = 0.45 and when x = 0.45, the regression equation predicts that y = 0.32 + 0.51 * 0.45 = 0.55. There is nothing special about the median here and you can obviously calculate the predicted value of Y for any X, for example when X= 0, the Y = 0.32 (the intercept) etc.

For any predicted value of Y there two two different prediction limits (confidence limits of the prediction). These can be calculated using the formulas in the text book or using R, but are difficult to calculate in JMP. If we are using JMP we will just estimate the prediction limits by eye.

One prediction limit is referred to as the prediction limits for the mean of Y and we can calculated 95%CL or 99%CL or whatever. So the 95%CL for the predicted mean of Y is the plausible range of values for the mean of the Y-values at a given X.

The other prediction limit is referred to as the prediction limits for a predicted value of Y. So the 95%CL for the predicted value of Y is the plausible range of values for the actual Y value predicted at a given X. This prediction limit is always larger than the prediction limits for the mean of Y.

To estimate these prediction limits in JMP, simply click on the red triangle beside Linear Fit and choose Confid Curves Fit (for the prediction limits of mean Y) and Confid Curves Indiv (for the prediction limits of the value of Y predicted by the regression line). I do not care which one you use, but be careful to be absolutely clear. Here they both are on the graph to the right, with the curved lines being the prediction limits of mean Y. By using the cross hair tool and zooming in on the graph I can make a pretty good estimate of the lower and upper prediction limits, as shown below for determining the lower limit (–0.0211) at X = 0.4475.

The whole thing is much easier in R (where input is green and output is blue, and the dataset is first loaded into R then made equal to dat just to simplify the coding) :

> dat=SalmonPaternity

> mod1=lm(y~x, data = dat) #create the regression model

> summary(mod1) #display the stats for the regression
Call:
lm(formula = y ~ x, data = dat)
Residuals:
Min 1Q Median 3Q Max
-0.52650 -0.23547 0.02081 0.18170 0.68347
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.31696 0.07917 4.003 0.000231 ***
x 0.51338 0.14949 3.434 0.001287 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2857 on 45 degrees of freedom
Multiple R-squared: 0.2077, Adjusted R-squared: 0.1901
F-statistic: 11.79 on 1 and 45 DF, p-value: 0.001287
> plot(y~x, data = dat) #plot the scatterplot
> abline(mod1) #add the regression line

> predy=data.frame(x=0.45)
> predict(mod1, predy, interval="predict") #this gives the prediction limits for a predicted value of Y for the given X
fit lwr upr
1 0.5479805 -0.03348593 1.129447
>
> predict(mod1, predy, interval="confidence") #this gives the prediction limits for a predicted mean of Y values for the given X
fit lwr upr
1 0.5479805 0.464053 0.6319081

Week 12: Regression

2013-11-23T11:08:00.001-05:00

Regression and correlation are married, or at least thinking about it. Correlation analysis is out on its own, without regression, but regression almost never goes out with correlation. Tortured analogies aside, you will see how these two forms of analysis provide different but related information about the relations between two variables

QUIZ: on-line, available at 3 pm on Monday and due (that means submitted by) 5 pm Tuesday. All of the questions are multiple choice and will be marked on-line. This week the quiz will be based mainly on chapter 16 in the textbook (on Correlation), but I reserve the right to ask questions about anything we have covered to date. I will try to remember to include a final comment box on the quiz this week.

HOMEWORK: This week's focus for study, practice, and assignment is Chapter 17 in the textbook on Regression. You should also read and think about the INTERLEAFs immediately before and after this chapter on Publication Bias, and Using Species as Data Points. This is the last chapter that we will cover in this course but you might find it interesting and informative to read the final few chapters as well, especially if you are thinking of going on in biology.

PRACTICE EXERCISES: chapter 17 problems 3, 9, and 11 (below the fold).

ASSIGNMENT: DUE before 11:30 on Friday 29 November, uploaded as PDF to Moodle and with the filename in the usual format.

PRACTICE EXERCISES

ASSIGNMENT

I am well aware that you all have a lot of work to do at the end of term, so I have tried to keep this assignment as straightforward as possible. I do, however, want to give you experience with, and information on, the various regression analyses that I asked you for on Project B. Thus there are many questions here but do try to answer them as briefly and succinctly as you can. Once you have done this assignment, that part of Project B will be easy (well, easier).

Using the Sleep dataset that we have been using for several assignments, analyse the relation between Non-dreaming and Dreaming portions of the sleep cycle, and answer the following questions, assuming that Dreaming is the response:

a. what is the equation that predicts the time spent in the Dreaming part of the sleep cycle from the time spent in the Non-dreaming part? What is the correlation between these two variables?

b. what is the median time spent Non-dreaming? For that median time, what is the predicted mean amount of time spent Dreaming, and what are the 95% confidence limits on that prediction [note that this calculation is easy in R, and nearly impossible in JMP; if you are using JMP plot the prediction interval on the graph and just estimate the confidence limits at the median value of X]

c. using the 95%CL of the slope and intercept of this relation, estimate whether the slope and intercept of the regression line are significantly different from zero? Interpret those results biologically (keep it simple).

d. is the relation between Dreaming and Non-dreaming time statistically significant? [report the test statistics, p-value etc.

e. plot the (Dreaming) residuals of this regression against the Non-dreaming time, as well as the normal quantile plot for those residuals, and assess whether you think the assumptions of this regression analysis have been met, and why you think that.

Correlation analysis

2013-11-20T21:08:00.003-05:00

Here are some basic procedures for correlation analysis in JMP and R, using the Candy Bar dataset I showed in class.

USING JMP

Step 1. Load up the dataset from Help>Sample Data >Food and Nutrition

Step 2. Choose Analyze>Multivariate Methods>Multivariate, and put all the variables you want to analyze into Y, Columns and click on OK

Step 3. If not visible, choose from under the red triangle: Correlations Multivariate, Pairwise Correlations, Nonparametric Correlations>Spearman's, and Scatterplot Matrix:

and get

IMPORTANT NOTE: if there are missing data, the correlation coefficients in the default correlation matrix (Correlations, above) will not match those under Pairwise Correlations, as shown below in the green boxes where the correlations between Cholesterol and Saturated Fat are different. That's because JMP uses a special algorithm to calculate correlations for this default correlation matrix when there are missing data, as indicated by the statement outlined in the RED rectangle below. DO NOT USE THIS METHOD as it is not in general use, and is unnecessarily confusing.

Instead, change the default method for the correlation matrix by choosing JMP>Preferences>Platform>Multivariate>Estimation Method>Pairwise as shown below, then close the preferences and rerun the correlation analysis as above.

The resulting correlation matrix (Correlations) will now have the same values as under Pairwise Correlations

Note that in the correlation matrix above, the positive correlations are shown in blue, negative in red, and significant correlations in bold font. The matrix itself can be turned into a data table in JMP by control-(right-) clicking on the matrix and choosing Make into Data Table, then that can be Exported as a text file and imported into your word processor.

USING R

Below I have put your input in green and R's output in blue

Step 1. Load the dataset into an object named dat (or whatever name you like)

dat=read.csv(file.choose())

Step 2. get the names of the variables

names (dat)

[1] "Brand" "Name" "Serving.pkg" "Oz.pkg" "Calories" "Total.fat.g" "Saturated.fat.g"

[8] "Cholesterol.g" "Sodium.mg" "Carbohydrate.g" "Dietary.fiber.g" "Sugars.g" "Protein.g" "Vitamin.A..RDI"

[15] "Vitamin.C..RDI" "Calcium..RDI" "Iron..RDI"

Step 3. get the correlation matrix for the parametric correlations

with(dat, cor(cbind(Saturated.fat.g, Cholesterol.g, Sodium.mg, Carbohydrate.g, Dietary.fiber.g, Sugars.g, Protein.g), use="complete.obs", method="pearson"))

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 1.00000000 0.40645869 -0.21226005 -0.02771014 0.45134428 0.12982705 0.3043595

Cholesterol.g 0.40645869 1.00000000 -0.09194795 -0.23675820 0.02222046 0.01909605 0.1500029

Sodium.mg -0.21226005 -0.09194795 1.00000000 0.05647324 -0.11342131 -0.18108132 0.2098704

Carbohydrate.g -0.02771014 -0.23675820 0.05647324 1.00000000 -0.01275351 0.82616981 -0.2610758

Dietary.fiber.g 0.45134428 0.02222046 -0.11342131 -0.01275351 1.00000000 -0.06920530 0.4397747

Sugars.g 0.12982705 0.01909605 -0.18108132 0.82616981 -0.06920530 1.00000000 -0.2951497

Protein.g 0.30435945 0.15000292 0.20987037 -0.26107578 0.43977471 -0.29514966 1.0000000

and for the nonparametric Spearman

with(dat, cor(cbind(Saturated.fat.g, Cholesterol.g, Sodium.mg, Carbohydrate.g, Dietary.fiber.g, Sugars.g, Protein.g), use="na.or.complete", method="spearman"))

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 1.000000000 0.44764285 -0.20551659 0.003443224 0.33965640 0.17981837 0.2815151

Cholesterol.g 0.447642847 1.00000000 -0.17185060 -0.211089179 0.09015895 0.04897635 0.1493562

Sodium.mg -0.205516590 -0.17185060 1.00000000 0.125961611 -0.04079726 -0.13156231 0.2273913

Carbohydrate.g 0.003443224 -0.21108918 0.12596161 1.000000000 -0.03642410 0.84364321 -0.3303399

Dietary.fiber.g 0.339656404 0.09015895 -0.04079726 -0.036424097 1.00000000 -0.09835628 0.4555465

Sugars.g 0.179818371 0.04897635 -0.13156231 0.843643207 -0.09835628 1.00000000 -0.3285223

Protein.g 0.281515132 0.14935620 0.22739135 -0.330339948 0.45554649 -0.32852233 1.0000000

NOTE that in both cases the term use="na.or.complete" says that you might have missing values and to go ahead anyway.

Step 4. To make a scatterplot matrix you will need to load the lattice package. The plot below shows the default version, but you can always change the colours and size etc if you know what you are doing. R does not plot the confidence ellipses as shown on the JMP plot (above)

library(lattice)

with(dat, splom(cbind(Saturated.fat.g, Cholesterol.g, Sodium.mg, Carbohydrate.g, Dietary.fiber.g, Sugars.g, Protein.g)))

Step 5. There is no easy way to get P-values with R so you have to install and load the Hmisc package first, then you can get both the correlation coefficients and the sample sizes and the P-values each in a separate matrix

library(Hmisc)

with(dat, rcorr(cbind(Saturated.fat.g, Cholesterol.g, Sodium.mg, Carbohydrate.g, Dietary.fiber.g, Sugars.g, Protein.g), type="pearson"))

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 1.00 0.41 -0.21 -0.03 0.45 0.13 0.30

Cholesterol.g 0.41 1.00 -0.04 -0.28 -0.08 0.05 0.01

Sodium.mg -0.21 -0.04 1.00 0.07 -0.14 -0.08 0.28

Carbohydrate.g -0.03 -0.28 0.07 1.00 -0.06 0.75 -0.23

Dietary.fiber.g 0.45 -0.08 -0.14 -0.06 1.00 -0.07 0.49

Sugars.g 0.13 0.05 -0.08 0.75 -0.07 1.00 -0.24

Protein.g 0.30 0.01 0.28 -0.23 0.49 -0.24 1.00

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 75 57 57 57 57 57 57

Cholesterol.g 57 75 75 75 75 75 75

Sodium.mg 57 75 75 75 75 75 75

Carbohydrate.g 57 75 75 75 75 75 75

Dietary.fiber.g 57 75 75 75 75 75 75

Sugars.g 57 75 75 75 75 75 75

Protein.g 57 75 75 75 75 75 75

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 0.0017 0.1129 0.8379 0.0004 0.3358 0.0213

Cholesterol.g 0.0017 0.7370 0.0150 0.4980 0.7014 0.8994

Sodium.mg 0.1129 0.7370 0.5355 0.2187 0.5152 0.0150

Carbohydrate.g 0.8379 0.0150 0.5355 0.5834 0.0000 0.0470

Dietary.fiber.g 0.0004 0.4980 0.2187 0.5834 0.5500 0.0000

Sugars.g 0.3358 0.7014 0.5152 0.0000 0.5500 0.0374

Protein.g 0.0213 0.8994 0.0150 0.0470 0.0000 0.0374

and the same for the Spearman

with(dat, rcorr(cbind(Saturated.fat.g, Cholesterol.g, Sodium.mg, Carbohydrate.g, Dietary.fiber.g, Sugars.g, Protein.g), type="spearman"))

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 1.00 0.45 -0.21 0.00 0.34 0.18 0.28

Cholesterol.g 0.45 1.00 -0.11 -0.28 0.00 0.08 0.06

Sodium.mg -0.21 -0.11 1.00 0.14 -0.13 -0.08 0.22

Carbohydrate.g 0.00 -0.28 0.14 1.00 -0.08 0.79 -0.28

Dietary.fiber.g 0.34 0.00 -0.13 -0.08 1.00 -0.09 0.52

Sugars.g 0.18 0.08 -0.08 0.79 -0.09 1.00 -0.30

Protein.g 0.28 0.06 0.22 -0.28 0.52 -0.30 1.00

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 75 57 57 57 57 57 57

Cholesterol.g 57 75 75 75 75 75 75

Sodium.mg 57 75 75 75 75 75 75

Carbohydrate.g 57 75 75 75 75 75 75

Dietary.fiber.g 57 75 75 75 75 75 75

Sugars.g 57 75 75 75 75 75 75

Protein.g 57 75 75 75 75 75 75

Saturated.fat.g Cholesterol.g Sodium.mg Carbohydrate.g Dietary.fiber.g Sugars.g Protein.g

Saturated.fat.g 0.0005 0.1251 0.9797 0.0097 0.1807 0.0339

Cholesterol.g 0.0005 0.3315 0.0165 0.9678 0.4920 0.6071

Sodium.mg 0.1251 0.3315 0.2451 0.2692 0.5037 0.0587

Carbohydrate.g 0.9797 0.0165 0.2451 0.5114 0.0000 0.0140

Dietary.fiber.g 0.0097 0.9678 0.2692 0.5114 0.4202 0.0000

Sugars.g 0.1807 0.4920 0.5037 0.0000 0.4202 0.0101

Protein.g 0.0339 0.6071 0.0587 0.0140 0.0000 0.0101

NOTE that even though using the Hmisc package requires a bit of extra preparation, the results are very easy to get all at once and can be copied and pasted right into your word processor

Repeatability

2013-11-18T21:56:00.002-05:00

On page 415 in the textbook the formula for Repeatability is given as:

where

These values can be taken from a typical ANOVA table of results:

as calculated from X=specimen identification (a letter, name or number to identify each individual that was measured more than once
and Y = the measurement

In the ANOVA table: Mean Square for specimen is MSgroups and Mean Square for Error is MSerror. The number of measurements for each specimen (individual) is n.

THIS FORMULA IS CORRECT and the one I presented in lecture last week was not. I have now corrected that on the pdf I posted of those lecture slides. My apologies for this error, and thanks to the student who pointed this out to me in class today.

Quiz Week 10

2013-11-18T21:05:00.000-05:00

Posted on Moodle. Must be answered on-line before 5 pm Tuesday 18 November.

This quiz is all about ANOVA and there are 10 questions spread over 2 different pages on-line. There are no datasets or calculations to be made. Just questions about some aspects of ASNOVA and interpreting the output from analyses.

There will be one last quiz next Monday, focussing on correlation but with at least half the questions from the rest of the course material.

Week 11: Correlation

2013-11-16T17:05:00.000-05:00

Correlation is an interesting topic as it looks at the relation between two continuous variables, one of which is usually designated as the predictor and one as the response, though sometimes it is hard (or impossible to tell which causes which. Remember, too, that correlation is not necessarily causation.

Despite my claims and best efforts, last week's assignment was hard, and a ton of work. I will try to compensate this week and next with much more straightforward assignments.

QUIZ: on-line, available at 3 pm on Monday and due (that means submitted by) 5 pm Tuesday. All of the questions are multiple choice and will be marked on-line. This week the quiz will be based mainly on chapter 15 in the textbook (on Analysis of Variance), but I reserve the right to ask questions about anything we have covered to date.

HOMEWORK: This week's focus for study, practice, and assignment is Chapter 16 in the textbook on Correlation. You should also read and think about the INTERLEAF on pages 429-430 about the sources of error in experimental design and statistical analysis—we have talked about this in class. Notice that we also talked about the 2D:4D ratios in their example 16.1.

PRACTICE EXERCISES: chapter 16 problems 1, 3, 8, and 11 (below the fold). I will almost certainly ask a version of question 1 on next week's quiz.

ASSIGNMENT: DUE before 11:30 on Friday 22 November, uploaded as PDF to Moodle and with the filename in the usual format.

PRACTICE EXERCISES (chapter 16)

ASSIGNMENT

I. Using the by-now familiar—and maybe even boring—dataset on the sleeping of mammals (Sleep dataset on Moodle for the week of 30 September), answer the following questions. In each case assume that I am a client who has come to you for statistical advice. I have collected these data and have hired you to analyze those data and to interpret my findings for me. I am interested in all of the relations between a couple of morphological variables (brain mass and body mass) and three estimates of time spent sleeping in different states on average per day (dreaming, non-dreaming, and total amount of sleep).

a. I looked at the correlations between all pairs of these variables in a correlation matrix and I see that for the parametric analyses all but two of the correlations are significant, but in the nonparametric analysis they are all significant. Why the difference? Provide me with a correlation matrix showing all possible correlations of these 5 variables, with parametric correlation coefficients above the diagonal and nonparametric below the diagonal, and all significant correlations (P<0.05) highlighted in bold font.

b. For one of the significant parametric correlations between a sleep variable and a morphological variable, do a thorough correlation analysis (plotting the data, assessing whether the assumptions have been met, checking for outliers and other anomalies, calculating the approximate 95%CL of the correlation coefficient, and assessing the significance of the correlation) and provide an interpretation of both the statistics and the biology.

c. I see that the correlation between Total Sleep and Body mass is negative. What does that mean?

Lecture slides and df

2013-11-15T14:04:00.001-05:00

I have posted my lectures slides from Wednesday and Friday as pdfs on Moodle.

DF
These slides show the correct calculation of df, unlike what I said in class on Friday.

Thus for ANOVA, df = (number of groups - 1), (total sample size - number of groups)

ANOVA in JMP and R

2013-11-13T21:24:00.004-05:00

Here are some tips on doing one factor (one way, single factor) ANOVA using either JMP or R. For these examples I am using the cholesterol.jmp dataset that is in the Sample Data in JMP (Help menu).

This dataset has two drug treatments, a placebo and a control and we will analyse the cholesterol levels for June PM (the final column of data.

USING JMP
Step 1. Make sure that the treatments (levels) for the predictor are all listed in one column and that column is a categorical variable (blue arrow below).

Step 2. Choose Analyze>Fit Y by X, and put the predictor (treatments) into X, Factor and the response into Y, Response, and get:

The blue box indicates the F-statistic and the P-value, the green box indicates the relevant df and the red box indicates the means for each level of the predictor (factor).

Report "There is significant variation in cholesterol levels among the 4 treatments (ANOVA, F= 222.6, df = 3,16, P <0.0001)"

You can see from the graph that A and B are similar and the Control and Placebo are similar, and you might estimate that they are not significantly different because the 95%CLs overlap

Step 3. Because the ANOVA is significant, you can now run a post hoc test to find out which pairs are sig diff. To do that choose Compare Means>All pairs, Tukey HSD from under the red triangle and get:

The green box indicates the relevant info. To make a graph like I showed in class you will have to add the letters by hand.

Step 4. You might want to check assumptions by doing quantile plots of each level to check for normality. You can also check for equal variances by choosing Unequal Variances from the menu under the red triangle and get:

The green arrows show that three of the test are significant, suggesting that the null hypothesis (equal variances) should be rejected. Also the rule of thumb that the largest SD should be no more than twice as big as the smallest is violated (green box). Because the variances are unequal, JMP does a Welch's ANOVA to correct for that (and the P-value is the same).

Step 5. Because the variances are so different and the sample size is very small you might also want to run a nonparametric ANOVA, by choosing Nonparametric>Wilcoxon test under the red triangle and get:

Where the red box indicates the appropriate test statistic and P-value. The results are the same, significant difference among mean cholesterols. To find out which pairs are sig diff, choose Nonparametric> Nonparametric multiple Comparisons>Wilcoxon Each Pair under the red triangle. I'll leave that for you to interpret on your own.

USING R
Same dataset and analyses as above. Here's the relevant script:

dat=read.csv(file.choose()) #read in the csv file and store in dat
 
boxplot(dat$June.PM~dat$treatment) #plot the data
 
mod1=aov(June.PM~treatment, data=dat) #one factor anova model
summary(mod1) #show the anova table
 
plot(mod1) #some diagnostic plots (complicated)
 
TukeyHSD(mod1) #Tukey posthoc test
 
kruskal.test(June.PM~treatment, data=dat) #nonparametric test

I will post again tomorrow on this R code and what it all means, as well as how rot interpret the results.

Week 10: ANOVA

2013-11-09T11:27:00.002-05:00

ANOVA (analysis of variance) is a vast topic so we are going to just barely scratch the surface in this course. If it interests you, BIOL-343, which will be offered for the first time in the fall of 2014, will focus entirely on ANOVA, or as it is more generally called, the General Linear Model.

In port because this past week covered so much and had a big assignment, this week's assignment is easy-peasy, and I will provide both JMP instructions and R code after the Wednesday lecture

QUIZ: on-line, available right after class on Monday and due (that means submitted by) 5 pm Tuesday. All of the questions are multiple choice and will be marked on-line. More details in a later post. This week the quiz will be based mainly on chapters 10-11 in the textbook, but I reserve the right to ask questions about anything we have covered to date.

HOMEWORK: This week's focus for study, practice, and assignment are chapters Chapter 14 and 15 in the textbook. You should also read and think about the INTERLEAF on pages 390-392. Chapter 14 is about the principles of experimental design—these are principles that are important to ensure that you can properly analyse the results of an experiment with the best statistical tools available. Chapter 15 is about ANOVA, a method for comparing more than two means, and is the basis for the analysis of most experimental results

PRACTICE EXERCISES: chapter 14 problems 8, 9 and 11, PLUS chapter 15 problems 1, 2, 3 and 5. See below the fold.

ASSIGNMENT: DUE before 11:30 on Friday 15 November, uploaded as PDF to Moodle and with the filename in the usual format.

PRACTICE EXERCISES

Chapter 14

Chapter 15

ASSIGNMENT

1. Using the Sleep dataset from the week of 30 September, available on Moodle, answer the questions posed below.

In this dataset the researchers have scored predation, exposure and danger each on a 1-5 scale where 1 is low and 5 is high. Thus predation refers to how much predation occurs on this species in nature, exposure refers to how exposed they are when they sleep (5 sleeps in the open, 1 in well-protected shelter), and danger refers to how much danger they are in over all when they are sleeping, so a species with lots of predation pressure, but sleeping in a well protected place might have a low danger level.

The researchers have hired you because of your prowess with statistics and want to know whether danger has any effect on the average number of hours spent sleeping per day (TotalSleep), the lifespan (LifeSpan, measured in years), and the gestation period (Gestation, measured in days) They have come to you for help with the analysis and interpretation of results.

a. Using parametric methods to answer the question: determine whether the assumptions of this method are satisfied; based on these methods, provide the researchers with an answer to their question, including an assessment of the validity of your answer (and the accompanying stats) and an analysis to compare each pair of danger levels (e.g. is the total sleep of mammals at danger level 1 different from that at danger level 5, etc.).

b. Using nonparametric methods to answer the question: provide the researchers with an answer to their question, including an assessment of the validity of your answer (and the accompanying stats) and an analysis to compare each pair of danger levels (e.g. is the total sleep of mammals at danger level 1 different from that at danger level 5, etc.)

c. Explain whether you would actually report your answer from a or b to your clients.

NOTE: some of you have asked whether a graph is necessary when you answer questions like this, because no graph was explicitly asked for. In my opinion, a graph almost always helps the reader to understand your work, by presenting the raw data and some summary stats where possible. Thus it's hard for me to imagine a situation where a graph would not be very useful. In that sense I guess you could say that a graph is always required.

Infographic of the week

2013-11-08T22:29:00.001-05:00

See below the fold

R example

2013-11-07T21:16:00.002-05:00

Here is a full R example using the Fitness dataset, exported from JMP as a .csv file, to compare the pulse rates measured at maximum levels and when running for a bunch of people. I will type the code in orange text and the output is in blue except where include pictures of the output so I can annotate and explain what things mean.

Step 1. Get the data into R (I always store the data in an object called dat, as this makes for efficiency in re-using code—you can use any name you want except data):

dat=read.csv(file.choose())

then I look at the column names just to make sure I am using the right names later:
names(dat)
[1] "Name" "Sex" "Age" "Weight" "Oxy" "Runtime" "RunPulse" "RstPulse" "MaxPulse"

Step 2. label the variables of interest to make running the code simpler

y1 = dat$MaxPulse
y2 = dat$RunPulse

Step 3. Plot the data to look for anomalies and see what's what:

boxplot(y1, y2, notch=TRUE)

These are notched box plots that also show the 95%CL of the median as the notch. It looks like the data are OK, and that MaxPulse > RunPulse when comparing the means

Step 4. Run the tests from yesterday's post that allow you to have the data in separate columns as they are here:

# independent 2-group t-test

t.test(y1,y2) # where y1 and y2 are numeric

# paired t-test

t.test(y1,y2,paired=TRUE) # where y1 & y2 are numeric

# independent 2-group Mann-Whitney U Test

wilcox.test(y1,y2) # where y and x are numeric

# dependent 2-group Wilcoxon Signed Rank Test

wilcox.test(y1,y2,paired=TRUE) # where y1 and y2 are numeric

Note that I didn't need all those comments (lines starting with #) but it was easiest just to copy the whole thing and paste it into R

and I got:

for the unpaired t-test

t.test(y1,y2) # where y1 and y2 are numeric

Welch Two Sample t-test

data: y1 and y2

t = 1.6719, df = 59.26, p-value = 0.09982

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

-0.8123965 9.0704610

sample estimates:

mean of x mean of y

173.7742 169.6452

for the paired t-test:

t.test(y1,y2,paired=TRUE) # where y1 & y2 are numeric

Paired t-test

data: y1 and y2

t = 6.0619, df = 30, p-value = 1.173e-06

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

2.737945 5.520120

sample estimates:

mean of the differences

4.129032

for the unpaired nonparametric test (Wilcoxon test):

wilcox.test(y1,y2) # where y and x are numeric

Wilcoxon rank sum test with continuity correction

data: y1 and y2

W = 587.5, p-value = 0.1325

alternative hypothesis: true location shift is not equal to 0

Warning message:

In wilcox.test.default(y1, y2) : cannot compute exact p-value with ties

That output (above) is just warning you that there are ties in your data and the p-value will be a somewhat inaccurate.

and for the paired nonparametric test

wilcox.test(y1,y2,paired=TRUE) # where y1 and y2 are numeric

Wilcoxon signed rank test with continuity correction

data: y1 and y2

V = 325, p-value = 1.139e-05

alternative hypothesis: true location shift is not equal to 0

Warning messages:

1: In wilcox.test.default(y1, y2, paired = TRUE) :

cannot compute exact p-value with ties

2: In wilcox.test.default(y1, y2, paired = TRUE) :

cannot compute exact p-value with zeroes

Again, the warnings are just to notify you that the p-value will be somewhat inaccurate.

Finally it might be useful to look at the distribution of the paired differences:
y3=y1-y2
boxplot(y3, notch=TRUE)

Above I have calculated a new variable, y3, by subtracting y2 from y1. In this notation, R is subtracting one vector (column) from the other at each row of the data. The result is the difference between y1 and y2 for each person and the box plot, below plots the distribution of those differences, which you can see is not zero, and since the 95%CL does not overlap with zero we can say that there is strong evidence that the difference between MaxPulse and Run Pulse is not zero (i.e., that they are significantly different).

NOTE: if you followed along through this R exercise you might realise that you can just set up whatever file you like as dat, then copy and paste all of the code above to do all of the required analyses and get the two (fairly nice) plots.

Two sample comparisons

2013-11-06T21:00:00.000-05:00

Using R and JMP to do some two sample comparisons, using the Fitness sample data set from JMP to compare the maximum and running pulse rates of 50 people.

USING JMP
Paired tests
select Analyze>Matched Pairs and put the columns of matched (paired) data into Y, Paired Response

select from the red triangle Plot Dif by Mean and Wilcoxon Signed Rank

The important bits are indicated with blue arrows: two-tailed P values, test statistics (t and S), sample size (N), and the actual difference between the paired means.

Unpaired Comparisons

To do this you have to have all the data in one column, and a separate column that labels which is which:

This time, select Analyze>Fit Y by X, and put the response (data) in Y, Response and the predictor (label) in X, Factor

Select from the red triangle Means/Anova/Pooled t and t Test

The t-test under Oneway Anova assumes that the variances of RunPulse and MaxPulse are equal, whereas under t Test it assumes they are not equal.

NOTE: ignore the sign of the t-statistic and don't bother putting a negative sign in front of it when you report it: t = 1.7, in this case

You can test whether the variances are equal by selecting Unequal Variances under the red triangle:

and you can see above that all 5 different tests for equality of variances are not significant (P>>0.05) so that means the variances are not sig diff (so they are equal)

To get the nonparametric test select Nonparametric>Wilcoxon Test under the red triangle:

this shows S = 869.5, P = 0.13, so not sig diff

USING R

Here is some useful R Code for the same dataset as above, with RunPulse (y1, below) and MaxPulse (y2, below) in separate columns. I will do these analyses and post results tomorrow, so you can see exactly what to do and how to interpret what you get. If you define y1 and y2 correctly, you just have to copy the code below and paste into R and hit return to get everything at once (one of the beauties of R.

# independent 2-group t-test
t.test(y~x) # where y is numeric and x is a binary factor
 
# independent 2-group t-test
t.test(y1,y2) # where y1 and y2 are numeric
 
# paired t-test
t.test(y1,y2,paired=TRUE) # where y1 & y2 are numeric
 
# independent 2-group Mann-Whitney U Test 
wilcox.test(y~A)  # where y is numeric and A is A binary factor
 
# independent 2-group Mann-Whitney U Test
wilcox.test(y,x) # where y and x are numeric
 
# dependent 2-group Wilcoxon Signed Rank Test 
wilcox.test(y1,y2,paired=TRUE) # where y1 and y2 are numeric

QUIZZES

2013-11-04T18:47:00.001-05:00

For the rest of this course, Monday quizzes will all be online, on Moodle. I am just learning the ropes of on-line quizzes so today's is a bit lame but next week's will be awesome.

Quizzes will be made available to you at 2:30 pm on Monday's right after class, and you will be allowed 30 min to answer all the questions. Any questions answered in that 30 min period will be automatically saved by Moodle. The deadline is 5 pm Tuesdays so you must complete the quiz before that time as access will be denied thereafter.

I would strongly recommend that you download any datasets associated with the quiz and load them up into your favourite stats analysis program before you begin the quiz.

Marks for each quiz should be available the next day, along with a list of the correct answers.

To provide you with a chance to test yourself on the material from the final week of the course, I may provide a quiz on the Monday following the last class, but I will give you details during that final week.

Starting to Work with R

2013-11-03T14:26:00.002-05:00

R is an excellent program for data analysis and statistical computing and we are going to try to introduce you to some of its features in the remainder of the course. R is particularly useful because:

it's free, open source, always available
it has a huge user and developer base so it's always up-to-date and getting better and more useful every day
it is THE statistical program for statisticians and data analysts everywhere
it allows you to keep a continuous record of your work, so you can always go back and repeat/.evaluate exactly what you have done
it makes top-notch graphs
there is a ton of info available on-line, including help sites, blogs, video tutorials and on-line analysis tools using R; there are also dozens of books available including a pretty good and relatively new "R For Dummies" and "Getting Started with R: an Introduction for Biologists"

To get started with R, do the following:

go to the R Project site; download and install the latest version of R for your operating system
go to the RStudio site and install that too (note that if you have the latest Mac OS, called Mavericks, you need to install a special version of RStudio, available here
run Rstudio and you will see that it is a shell for R that lets you run R without actually opening R itself. You can just run R alone if you want but I find the RSudio environment more useful
play around with RStudio to see how it works and all the options available. Type 4+3 into the console and hit return. R is a very useful and powerful calculator
check out the following useful websites

Quick-R: lots of useful instructions and scripts for all the analyses we will do in this course
R tutorial: this is a bit out of date but useful for beginners
Kickstarting R: lots of useful stuff
Introducing R: an online book
Task Views: advanced, but interesting listing of many of the kinds of things you can do with R
R-bloggers: tons of links to blog posts on R, most of which are about fairly advanced topics well beyond the scope of this course; worth looking at to see how versatile R is

To see some examples of cool graphs that you can plot in R, type demo(graphics) in the R console window

What lies ahead

2013-11-03T12:13:00.001-05:00

We will be making some changes to the quiz structure this week, but the assignments will continue as usual until the end of term. So this is what lies ahead that you will be marked on:

QUIZ: 4 more, one each Monday
ASSIGNMENTS: 4 more, one due each Friday
PROJECT B: due a week after the end of classes

Our learning schedule will follow the textbook, 'skipping' (or at least skimming) some chapters. Here's what we are going to focus on:

Week 9 (this week): two sample comparisons, chapters 12 and 13 in the textbook, plus the interleaf on pp 315-317. We are skipping/skimming over chapters 10 and 11.
Week 10: comparing more than two groups, textbook chapter 15, with a bit from chapter 14 on the design of experiments
Week 11: correlation, textbook chapter 16, plus interleaf 9 on experimental and statistical mistakes
Week 12: regression, textbook chapter 17.

Project B covers all of this material but you should be able to get started with data collection right away and keep analysing data as we go. Don't forget about using the tutorial sessions to get help.

Week 9: Comparing two samples, and R

2013-11-02T11:53:00.004-04:00

This week we begin the final third of the course in which we will explore statistics tests of (mostly) continuously distributed data, where the past two weeks we have focused on tests of counts and frequencies. We will also dip our toes in the vast lake of R.

QUIZ: in Monday's class, I will introduce a new type of quiz that we will probably use for the rest of the course. This week it will be based mainly on chapter 9 in the textbook, but for the rest of the term will cover the entire course so far.

HOMEWORK: We are going to skip over doing problems and assignments based on chapters 10 and 11 in the text but you should read them over so that you have at least a vague appreciation of that material because it forms the basis for what we will be doing for the rest of the term.

This week's focus for study, practice, and assignment are chapters Chapter 12 and 13 in the textbook. You should also read, study, understand, and have a good working knowledge of the INTERLEAF on pages 315-317. I will provide a handout that summarises some of this interleaf material. Chapters 12 and 13 take us into the worlds of parametric and nonparametric statistics by covering the myriad ways that you can compare two samples.

PRACTICE EXERCISES: chapter 12, problems 9, 11 and 13 PLUS chapter 13 problems 12 and 14. See below the fold.

ASSIGNMENT: DUE before 11:30 on Friday 8 November, uploaded as PDF to Moodle and with the filename in the usual format.

PRACTICE EXERCISES

Chapter 12

Chapter 13

ASSIGNMENT

NOTE: in each case, be sure you inspect the data before analysing it, and report the results of your analyses using the standard reporting method described in an earlier post. I am well aware that this is a big assignment, and I will provide instruction in class on Monday and Wednesday as well as various details here on the blog.

I. Using the Sleep dataset from Assignment 3, available on Moodle for the week of 30 September. To do this analysis you need to ensure that Predation levels are coded as Nominal and NOT a Continuous variable, whether you are doing this analysis in JMP or R, or whatever.
a. compare the average amount sleep (TotalSleep) obtained by mammals with Predation levels 2 (low) and 5 (high). Provide a biological interpretation of your results, as well as a statistical interpretation.
b. compare the amount of sleep obtained by mammals with Predation levels 1 (lowest) and 5 (highest). Provide a biological interpretation of your results, as well as a statical interpretation.
c. Briefly compare your findings from a and b

II. using that same sleep dataset, compare the average amount of time spent dreaming and not dreaming. Note that you have data for these two variables for each species so you can do a paired comparison.
a. report the results and interpretation of parametric and nonparametric tests.
b. Which test is best for these data, and why?

Male House Finch

III. In the textbook chapter 13 Assignment question 16 (see below), an experiment is described with zebra finches. We did exactly the same experiment with house finches and got the data in the file hofi.csv on Moodle.

a. does the mean PHA differ between the birds that received carotenoid supplements and those that did not get the supplements?
b. did mean SRBC differ between those two groups?
c. do both parametric and nonparametric tests for a and b and compare your results. Explain why the results are different, or why they are not different.

Project B

2013-11-01T20:53:00.002-04:00

For this second Project, collect the same data from three different populations. For each ’individual’ obtain data for two different traits that vary continuously and that you have good reason to think might be related, positively or negatively. You can either measure things yourself, or take data from the web. Whatever you decide to study you must sample the populations rather than taking all of the data available. See examples below but you are not allowed to use these particular datasets.

Answer the following questions about the data collected:

for each trait, do the mean values differ between populations?
within each population is there any relation between the two traits? What is the equation to predict one trait from the other in each population (even if the relation is not statistically significant)?
do the slopes of the relations between the two traits differ between populations?
what is the predicted value (+/- 95%CL) of the response variable at the median of the predictor variable for each of the relations between response and predictor (for each population)?
what do you conclude from this study

Write a brief report in the typical format for a scientific paper: title, abstract, introduction, methods, results, discussion, and references (if any). You do not have to answer the questions above in any particular order, just collect and analyse your data so that you can address those questions in your Results. The abstract and intro need only be be 2-3 sentences each. The data and analyses should be summarized in graphs and tables as much as possible. You must include graphs showing the means and 95%CL for each trait in each population, organized to facilitate comparisons, plus scatter plots of the response and predictor for each population, including the regression lines. Include references if these will help to make the discussion more useful include enough details in your methods that someone else could repeat your study and analyses.

Submit on Moodle as a PDF plus a .csv file of all the raw data before noon on 6 December 2013. Your mark will be based in the clarity of your report, the correctness of your analyses, the quality of your graphs, and the thoughtfulness of your Discussion.

EXAMPLES
1. Get data on the salary and the number of years of experience of players in professional soccer, with data from the English premier league, the Spanish league and the Italian league. There are lots of plays so I might be inclined to use a stratified random sampling, randomly sampling 5 players from each team.

2. Measure the length of leaf and petiole on the leaves of 3 different but closely related tree species. I would sample haphazardly taking one leaf from each of 25 different trees scattered around Kingston.

3. Measure the relation between basal metabolic rate and body mass in rodents, carnivores, and ungulates (data available online). Randomly choose 25 species from each group for analysis.

Contingency Table Analysis in JMP

2013-10-30T20:55:00.004-04:00

In JMP, it would be tempting to set up a 2x2 table like this

but that would not work. The correct setup is with two columns of categorical data and one of counts

Then choose Fit Y by X and set up the dialog box like this

and you will get this

use the red triangle beside Contingency Table to get further options.