LAB EXERCISE A: Correlation Analysis (Pearson's r)
Like Chi-squared, correlation analysis determines whether two variables are associated with one another, whether positively or negatively. Unlike Chi-squared, simple linear correlation (Pearson's r) relates two variables measured at the interval or ratio level. Also, not only can Pearson's r tell you IF a relationship exists (like Chi-squared) but how STRONG that association is (Chi-squared requires doing a Yule's Q to get this information, which you can only use on a 2 x 2 table). The Pearson's r co-efficient ranges from -1 to +1, where a perfect, all-lined-up, direct relationship shows a co-efficient of +1 and a perfect inverse relationship shows a co-efficient of -1. The closer to 0 the co-efficient is, the more random the association.
Ideally, you can get started by simply graphing your two variables against one another on a common X-Y scattergram. For your first pair of data, you'd look up the X variable on the X axis and read straight up until you come to the height of the Y variable on the Y axis. At that intersection, you mark a point. You keep on going for as many X-Y pairs as you have. Examination of the scattergram can tell you a lot about the relationship: its direction, its strength, and its linearity. If the cloud of dots trends to the upper right of the graph, you have a direct relationship; if it trends down to the lower right, you have an inverse relationship. If the cloud is very narrow, like a cigar, it's a strong relationship; it's weaker if it looks more like a football; there is basically no discernible relationship if the cloud looks more like a soccer ball. If the relationship does appear to exist on the basis of this cloud's shape, examine it more closely to see if its longest axis falls roughly along a line. Pearson's r is for linearly related data: If the cloud has a bent shape, like a J or an L, it must be transformed mathematically to force it into a straight line so that Pearson's r will work with it (very commonly, you can linearize such data with logarithmic transformations). If the cloud is U shaped, though, you're out of luck. With the 10 cases we have in this exercise (professorial kindness), we are unlikely to detect these sorts of problems (non-linearity, heteroscedasticity), so we'll plow forward to illustrate the process of performing a simple correlation and regression analysis. Lucky you.
Below is a spreadsheet summarizing student performance and a variety of class and instructor attributes for ten randomly chosen sample sections of a GIS seminar given at the various Nationwide University campuses in California's industrial and office complexes. Each class section had ten students enrolled. How convenient the math!
The first column is the location of the classes. The second is average student performance for each of the ten class sections. These scores have been converted into a 1-5 point system (grades on a 1-5 scale, with 5 being best, an A, and 1 being worst, an F). The third column is the average number of hours spent by students in outside study and project preparation each week in each of the sections (based, of course, on student self-reporting, which may be a bit self-serving!). The fourth column is student perception of the ten instructors' receptiveness to students and willingness to provide individualized attention and support. The fifth column consists of student ratings of the courses in terms of the organization and clarity of the instructors and materials provided. The last column is the average number of years that students have spent working for their present employers (Nationwide University targets its classes to working professionals, managers, and technical people in industry).
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COURSE STUDENT OUTSIDE INSTR- ORGANI- WORK EX-
LOCATION GRADES STUDY SUPPORT ZATION PERIENCE
Y X1 X2 X3 X4
Bakersfield 3.638 2.154 4.154 3.369 4.338
Fresno 4.018 2.682 3.288 3.676 3.776
Irvine 4.241 2.193 4.367 3.671 4.133
Los Angeles 4.278 2.719 3.931 4.147 4.378
Riverside 3.615 2.366 3.949 3.163 3.956
Sacramento 4.200 2.500 3.900 3.700 4.200
San Diego 4.184 1.805 3.992 4.119 4.168
San Francisco 4.040 3.071 3.494 2.849 3.877
San Jose 4.011 2.310 4.182 3.372 3.787
Van Nuys 3.932 2.511 4.063 2.995 3.842
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Now, just because I'm a semi-compassionate person (I'd score pretty highly on
X2?), I have also made this spreadsheet available to you on the web. Make
sure you're on a PC with both Netscape (or similar graphical browser) and
Excel. Click on this link, http://www.csuchico.edu/~lapaloma/geog215/215lab9.xls,
and either Netscape will prompt you to save the file somewhere or will
directly evoke Excel and plop you right in the spreadsheet. If this doesn't
work, you can manually type it into your spreadsheet.
It's about time you became acquainted with the statistical functions of your spreadsheet (Excel for most, if not all, of you). Up until now, I've been having you calculate everything in terms of columns and rows, to mitigate the math phobia so many students have. Now, it's time to find out some other nifty things spreadsheets can do these days (and this should speed things up for you a lot). For each variable, calculate the indicated statistics, using these statistical functions.
You can do this a couple of different ways. First, you can type in the function, e.g., =stdev(b4:b13), in the cell where you want the answer to land. Alternatively, you can highlight that cell and click on the "fx" button on top ("function 'wizard'") and then choose the "statistical" function on the left menu and then pick whatever function you want to perform on the right menu that comes up (e.g., stdev for samples or stdevp for populations). A dialogue box will come up for you to input the range, on which you want the math performed (e.g., b4:b13). Hit "finish," and there you have it.
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see below STUDENT OUTSIDE INSTR- ORGANI- WORK EX-
for Excel GRADES STUDY SUPPORT ZATION PERIENCE
definitions Y X1 X2 X3 X4
minima _______ _______ _______ _______ _______
maxima _______ _______ _______ _______ _______
means _______ _______ _______ _______ _______
medians _______ _______ _______ _______ _______
std dev _______ _______ _______ _______ _______
variances _______ _______ _______ _______ _______
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Excel formulas for the cells in which you'll put your results:
minima =min(b4:b13) or whatever your range is
maxima =max(b4:b13)
means =average(b4:b13)
medians =median(b4:b13)
std dev =stdev(b4:b13)
variances =var(b4:b13)
Punctuation is very important!
And don't forget you can block copy one formula across the row!
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r t prob
student grades and outside study _______ _______ _______
work experience and outside study _______ _______ _______
instructor support and work experience _______ _______ _______
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Excel formulas for the cells in which you're going to put your results:
r =correl(c4:c13,b4:b13)
t =(b16*(sqrt(10-2)))/sqrt(1-b16^2)
b16 is used here to mean whichever cell you put r in, and you may
well have designed your spreadsheet to put it in some other cell
Again, punctuation is critical and you can block copy to save typing.
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LAB EXERCISE B: Simple Linear Regression Analysis
Unlike correlation, regression tests for causal importance and models the degree of influence that one independent variable, X, has over another dependent variable, Y. That is, it measures the degree to which variation in X explains variation in Y. You can come up with such statements as "Variable 1 explains 47 percent of the variation in Variable 2."
This means that you have to set up hypotheses expressing your sense of the direction of causality. These hunches are grounded in theory or in plain common sense where theory is not yet well-developed.
Let's use student performance as the dependent, Y, variable. Let us then try out each of the other variables as independent variables, X1, X2, X3, and X4, proposed as explanations for student performance. Let us further assume that NU is doing an exploratory study and would be more uncomfortable with a Type II error. In other words, let us choose a somewhat generous alpha, 0.10 for our testing standard.
Since, regression (unlike correlation) entails specifying an expected direction, we will be using a one-tailed test. A common test statistic for the co-efficient of determination (regression co-efficient) is F, which is designed for one-tailed tests. To enter an F table, you need to know WHICH table to enter (0.10) and two different figures for degrees of freedom (remember that bit from the ANOVA lab?). For a test of r2, DF1 is equal to k or the number of independent variables in any one test (in this case, 1 for each of the 4 tests) and DF2 is equal to the number of cases (10) minus k (1) minus 1, or 8 in all.
In plain English, please state your expectations of the relationships between each X and the single Y. State them formally as alternate or working hypotheses and as null hypotheses. For each alternate hypothesis, also state why you expect what you do.
Ha:
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Ho:
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Ho:
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Ha:
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Ho:
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Ho:
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r r2 b a F *?
X1-->Y _______ _______ _______ _______ _______ ___
X2-->Y _______ _______ _______ _______ _______ ___
X3-->Y _______ _______ _______ _______ _______ ___
X4-->Y _______ _______ _______ _______ _______ ___
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Excel formulas for regression elements
r =correl(b4:b13,c4:c13)
r2 =b16^2
b16 is used here to symbolize wherever you stuck your r calculation
b =slope(b4:b13,d4:d13)
It is very important that your Y variable be the first array listed
Follow the Y array by a comma and then the X array
a =intercept (b4:b13,e4:e13)
Again, the Y array has to come first
F =(b20*(10-2))/(1-b20)
b20 is used to symbolize wherever you put your r2
Punctuation is really important here, folks
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Y = a + bX
Y = _____ + _____X (don't forget the sign for b)
There, you've built what may be your first regression model(s)!
At this point, many students become concerned, saying they have calculated a
and b but they don't know how to calculate X and Y. Calm down: You've
created a model of the relationship between X and Y. What can you do with
this? You can select any value of X and use the model to predict what the
expected value of Y should be, if no other factors were muddling up this X-Y
interaction. You'll get to fiddle with this predictive function of simple
linear regression a little farther down.
Y
5 +---------+---------+---------+---------+---------+
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4 +---------+---------+---------+---------+---------+
s | | | | | |
t | | | | | |
u | | | | | |
d | | | | | |
e 3 +---------+---------+---------+---------+---------+
n | | | | | |
t | | | | | |
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g | | | | | |
r 2 +---------+---------+---------+---------+---------+
a | | | | | |
d | | | | | |
e | | | | | |
s | | | | | |
1 +---------+---------+---------+---------+---------+
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0 +---------+---------+---------+---------+---------+
0 1 2 3 4 5
X
Name of Xi Variable: __________________________________________
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first placed on the web: 11/27/98
last revised: 11/29/98
© Dr. Christine M.
Rodrigue