Principal Component Analysis (PCA) is a handy statistical tool to always have available in your data analysis tool belt.
It’s a data reduction technique, which means it’s a way of capturing the variance in many variables in a smaller, easier-to-work-with set of variables.
There are many, many details involved, though, so here are a few things to remember as you run your PCA.
In Part 3 and Part 4 we used the lm() command to perform least squares regressions. We saw how to check for non-linearity in our data by fitting polynomial models and checking whether they fit the data better than a linear model. Now let’s see how to fit an exponential model in R.
As before, we will use a data set of counts (atomic disintegration events that take place within a radiation source), taken with a Geiger counter at a nuclear plant.
The counts were registered over a 30 second period for a short-lived, man-made radioactive compound. We read in the data and subtract the background count of 623.4 counts per second in order to obtain
One important consideration in choosing a missing data approach is the missing data mechanism—different approaches have different assumptions about the mechanism.
Each of the three mechanisms describes one possible relationship between the propensity of data to be missing and values of the data, both missing and observed.
Missing Completely at Random, MCAR, means there is no relationship between (more…)
In all linear regression models, the intercept has the same definition: the mean of the response, Y, when all predictors, all X = 0.
But “when all X=0” has different implications, depending on the scale on which each X is measured and on which terms are included in the model.
So let’s specifically discuss the meaning of the intercept in some common models:
In Part 3 we used the lm() command to perform least squares regressions. In Part 4 we will look at more advanced aspects of regression models and see what R has to offer.
One way of checking for non-linearity in your data is to fit a polynomial model and check whether the polynomial model fits the data better than a linear model. However, you may also wish to fit a quadratic or higher model because you have reason to believe that the relationship between the variables is inherently polynomial in nature. Let’s see how to fit a quadratic model in R.
We will use a data set of counts of a variable that is decreasing over time. Cut and paste the following data into your R workspace.
A <- structure(list(Time = c(0, 1, 2, 4, 6, 8, 9, 10, 11, 12, 13,
14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30),
Counts = c(126.6, 101.8, 71.6, 101.6, 68.1, 62.9, 45.5, 41.9,
46.3, 34.1, 38.2, 41.7, 24.7, 41.5, 36.6, 19.6,
22.8, 29.6, 23.5, 15.3, 13.4, 26.8, 9.8, 18.8, 25.9, 19.3)), .Names = c("Time", "Counts"),
row.names = c(1L, 2L, 3L, 5L, 7L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 19L, 20L, 21L, 22L, 23L, 25L, 26L, 27L, 28L, 29L, 30L, 31L),
class = "data.frame")
Let’s attach the entire dataset so that we can refer to all variables directly by name.
attach(A)
names(A)
First, let’s set up a linear model, though really we should plot first and only then perform the regression.
linear.model <-lm(Counts ~ Time)
We now obtain detailed information on our regression through the summary() command.
summary(linear.model) Call: lm(formula = Counts ~ Time) Residuals: Min 1Q Median 3Q Max -20.084 -9.875 -1.882 8.494 39.445 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 87.1550 6.0186 14.481 2.33e-13 *** Time -2.8247 0.3318 -8.513 1.03e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 15.16 on 24 degrees of freedom Multiple R-squared: 0.7512, Adjusted R-squared: 0.7408 F-statistic: 72.47 on 1 and 24 DF, p-value: 1.033e-08
The model explains over 74% of the variance and has highly significant coefficients for the intercept and the independent variable and also a highly significant overall model p-value. However, let’s plot the counts over time and superpose our linear model.
plot(Time, Counts, pch=16, ylab = "Counts ", cex.lab = 1.3, col = "red" )
abline(lm(Counts ~ Time), col = "blue")
Here the syntax cex.lab = 1.3 produced axis labels of a nice size.
The model looks good, but we can see that the plot has curvature that is not explained well by a linear model. Now we fit a model that is quadratic in time. We create a variable called Time2 which is the square of the variable Time.
Time2 <- Time^2
quadratic.model <-lm(Counts ~ Time + Time2)
Note the syntax involved in fitting a linear model with two or more predictors. We include each predictor and put a plus sign between them.
summary(quadratic.model) Call: lm(formula = Counts ~ Time + Time2) Residuals: Min 1Q Median 3Q Max -24.2649 -4.9206 -0.9519 5.5860 18.7728 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 110.10749 5.48026 20.092 4.38e-16 *** Time -7.42253 0.80583 -9.211 3.52e-09 *** Time2 0.15061 0.02545 5.917 4.95e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 9.754 on 23 degrees of freedom Multiple R-squared: 0.9014, Adjusted R-squared: 0.8928 F-statistic: 105.1 on 2 and 23 DF, p-value: 2.701e-12
Our quadratic model is essentially a linear model in two variables, one of which is the square of the other. We see that however good the linear model was, a quadratic model performs even better, explaining an additional 15% of the variance. Now let’s plot the quadratic model by setting up a grid of time values running from 0 to 30 seconds in increments of 0.1s.
timevalues <- seq(0, 30, 0.1)
predictedcounts <- predict(quadratic.model,list(Time=timevalues, Time2=timevalues^2))
plot(Time, Counts, pch=16, xlab = "Time (s)", ylab = "Counts", cex.lab = 1.3, col = "blue")
Now we include the quadratic model to the plot using the lines() command.
lines(timevalues, predictedcounts, col = "darkgreen", lwd = 3)
The quadratic model appears to fit the data better than the linear model. We will look again at fitting curved models in our next blog post.
About the Author: David Lillis has taught R to many researchers and statisticians. His company, Sigma Statistics and Research Limited, provides both on-line instruction and face-to-face workshops on R, and coding services in R. David holds a doctorate in applied statistics.
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One of the nice features of SPSS is its ability to keep track of information on the variables themselves.
This includes variable labels, missing data codes, value labels, and variable formats. Spending the time to set up variable information makes data analysis much easier–you don’t have to keep looking up whether males are coded 1 or 0, for example.
And having them all in the variable view window makes things incredibly easy while you’re doing your analysis. But sometimes you need to just print them all out–to create a code book for another analyst or to include in the output you’re sending to a collaborator. Or even just to print them out for yourself for easy reference.
There is a nice little way to get a few tables with a list of all the variable metadata. It’s in the File menu. Simply choose Display Data File Information and Working File.
Doing this gives you two tables. The first includes the following information on the variables. I find the information I use the most are the labels and the missing data codes.
Even more useful, though, is the Value Label table.
It lists out the labels for all the values for each variable.
So you don’t have to remember that Job Category (jobcat) 1 is “Clerical,” 2 is “Custodial,” and 3 is “Managerial.”
It’s all right there.