Statics

BUS308 Week 4 Lecture 1

Our investigation changes focus a bit this week. We started the class by finding ways to describe and summarize data sets – finding measures of the center and dispersion of the data with means, medians, standard deviations, ranges, etc. As interesting as these clues were, they did not tell us all we needed to know to solve our question about equal work for equal pay. In fact, the evidence was somewhat contradictory depending upon what measure we focused on. In Weeks 2 and 3, we changed our focus to asking questions about differences and how important different sample outcomes were. We found that all differences were not important, and that for many relatively small result differences we could safely ignore them for decision making purposes – they were due to simple sampling (or chance) errors. We found that this idea of sampling error could extend into work and individual performance outcomes observed over time; and that over- reacting to such differences did not make much sense.

Now, in our continuing efforts to detect and uncover what the data is hiding from us, we want to start to find out why something happened, what caused the data to act as it did? This week we move from examining differences to looking at relationships; that is, if some measure changes does another measure change as well? And, if so, can we use this information to make predictions and/or understand what underlies this common movement?

Our tools in doing this involve correlation, the measurement of how closely two variables move together; and regression, an equation showing the impact of inputs on a final output. A regression is similar to a recipe for a cake or other food dish; take a bit of this and some of that, put them together, and we get our result.

Correlation

We have seen correlations a lot, and probably have even used them (formally or informally). We know, for example, that all other things being equal; the more we eat. the more we weigh. Kids, up to the early teens, grow taller the older they get. If we consistently speed, we will get more speeding tickets than those who obey the speed limit. The more efforts we put into studying, the better grades we get. All of these are examples of correlations.

Correlations exist in many forms. A somewhat specialized correlation was the Chi Square contingency test (for multi-row, multi-column tables) we looked at last week, if we find the distributions differ, then we say that the variables are related/correlated. This correlation would run from 0 (no correlation) thru positive values (the larger the value the stronger the relationship).

Probably the most commonly used correlation is the Pearson Correlation Coefficient, symbolized by r. It measures the strength of the association – the extent to which measures change together – between interval or ratio level measures. Excel’s Fx Correl, and the Data Analysis Correlation both produce Pearson Correlations.

Most correlations that we are familiar with show both the direction (direct or inverse) as well as the strength of the relationship, and run from -1.0 (a strong and perfect inverse correlation) through 0 (a weak and non-existent correlation) to +1.0 (a strong an perfect direct

correlation). A direct correlation is positive; that is, both variables move in the same direction, such as weight and height for kids. An inverse, or negative, correlation has variables moving in different directions. For example, the number of hours you sleep and how tired you feel; the more hours, the less tired while the fewer hours, the more tired.

The strength of a correlation is shown by the value (regardless of the sign). For example, a correlation of +.78 is just as strong as a correlation of -.78; the only difference is the direction of the change. If we graphed a +.78 correlation the data points would run from the lower left to the upper right and somewhat cluster around a line we could draw thru the middle of the data points. A graph of a -.78 correlation would have the data points starting in the upper left and run down to the lower right. They would also cluster around a line.

Correlations below an absolute value of around .70 are generally not considered to be very strong. The reason for this is due to the coefficient of determination(CD). This equals the square of the correlation and shows the amount of shared variation between the two variables. Shared variation can be roughly considered the reason that both variables move as they do when one changes. The more the shared variation, the more one variable can be used to predict the other. If we square .70 we get .49, or about 50% of the variation being shared. Anything less is too weak of a relationship to be of much help.

Students often feel that a correlation shows a “cause-and-effect” relationship; that is, changes in one thing “cause” changes in the other variable. In some cases, this is true – height and weight for pre-teens, weight and food consumption, etc. are all examples of possible cause- and- effect relationships; but we can argue that even with these there are other variables that might interfere with the outcomes. And, in research, we cannot say that one thing causes or explains another without having a strong correlation present.

However, just as our favorite detectives find what they think is a cause for someone to have committed the crime, only to find that the motive did not actually cause that person to commit the crime; a correlation does not prove cause-and-effect. An example of this is the example the author heard in a statistics class of a perfect +1.00 correlation found between the barrels of rum imported into the New England region of the United States between the years of 1790 and 1820 and the number of churches built each year. If this correlation showed a cause- and-effect relationship, what does it mean? Does rum drinking (the assumed result of importing rum) cause churches to be built? Does the building of churches cause the population to drink more rum?

As tempting as each of these explanations is, neither is reasonable – there is no theory or justification to assume either is true. This is a spurious correlation – one caused by some other, often unknown, factor. In this case, the culprit is population growth. During these years – many years before Carrie Nation’s crusade against Demon Rum – rum was the common drink for everyone. It was even served on the naval ships of most nations. And, as the population grew, so did the need for more rum. At the same time, churches in the region could only hold so many bodies (this was before mega-churches that held multiple services each Sunday); so, as the population got too large to fit into the existing churches, new ones were needed.

At times, when a correlation makes no sense we can find an underlying variable fairly easily with some thought. At other times, it is harder to figure out, and some experimentation is needed. The site http://www.tylervigen.com/spurious-correlations is an interesting website devoted to spurious correlations, take a look and see if you can explain them. 😊😊

Regression

Even if the correlation is spurious, we can often use the data in making predictions until we understand what the correlation is really showing us. This is what regression is all about. Earlier correlations between age, height, and even weight were mentioned. In pediatrician offices, doctors will often have charts showing typical weights and heights for children of different ages. These are the results of regressions, equations showing relationships. For example (and these values are made up for this example), a child’s height might be his/her initial height at birth plus and average growth of 3.5 inches per year. If the average height of a newborn child is about 19 inches, then the linear regression would be:

Height = 19 inches plus 3.5 inches * age in years, or in math symbols:

Y = a + b*x, where y stands for height, a is the intercept or initial value at age 0 (immediate birth), b is the rate of growth per year, and x is the age in years.

In both cases, we would read and interpret it the same way: the expected height of a child is 19 inches plus 3.5 inches times its age. For a 12-year old, this would be 19 + 3.5*12 = 19 + 42 = 61 inches or 5 feet 1 inch (assuming the made-up numbers are accurate).

That was an example of a linear regression having one output and a single, independent variable as an input. A multiple regression equation is quite similar but has several independent input variables. It could be considered to be similar to a recipe for a cake:

Cake = cake mix + 2* eggs + 1½ * cup milk + ½ * teaspoon vanilla + 2 tablespoons* butter.

A regression equation, either linear or multiple, shows us how “much” each factor is used in or influences the outcome. The math format of the multiple regression equation is quite similar to that of the linear regression, it just includes more variables:

Y = a + b1*X1 + b2*X2 + b3*X3 + …; where a is the intercept value when all the inputs are 0, the various b’s are the coefficients that are multiplied by each variable value, and the x’s are the values of each input.

A note on how to read the math symbols in the equations. The Y is considered the output or result, and is often called the dependent variable as its value depends on the other factors. The different b’s (b1, b2, etc.) are coefficients and read b-sub-1, b-sub-2, etc. The subscripts 1, 2, etc. are used to indicate the different coefficient values that are related to each of the input variables. The X-sub-1, X-sub-2, etc., are the different variables used to influence the output, and are called independent variables. In the recipe example, Y would be the quality of the cake, a would be the cake mix (a constant as we use all of what is in the box), the other ingredients would relate to the bX terms. The 2*eggs would relate to b1*X1, where b1 would equal 2 and X1 stands for eggs, the second input relates to the milk, etc.

If you have any questions on this material, please ask your instructor.

After finishing with this lecture, please go to the first discussion for the week, and engage in a discussion with others in the class over the first couple of days before reading the second lecture.