INDEPENDENT VARIABLES AND CHI SQUARE

Independent versus Dependent Variables Given two variables X and Y, they are said to be independent if the occurance of one does not affect the probability of the occurence of the other. Formally, X and Y are independent if P (X | Y) = P (X) or P (Y | X) = P (Y) What does it mean?

Independent versus Dependent Variables y 1 … y k … y q y 1 … y k … y q x1x1……xhxh……XpXpx1x1……xhxh……XpXp…… n n 1k … n 1q n n 1k … n 1q n h1 … n hk … n hq n h1 … n hk … n hq n p1 … n pk … n pq n p1 … n pk … n pq n 10 n h0 n p0 n 01 n 0k n pq n Consider the following contingency table X Y We say that X is independent from Y if

Independent versus Dependent Variables: example 1 The following table gives a contingency table of an observed population (in million) based on gender (X) and healt insurance coverage (Y). Are the two variables independent? That is the health insurance coverage depends on gender? Covered by healt insurance Not Covered by healt insurance Total Male Female Total

Independent versus Dependent Variables Covered by healt insurance Not Covered by healt insurance Total Male Female Total111 We have to verify 2. YES X Y

Independent versus Dependent Variables Covered by healt insurance Not Covered by healt insurance Total Male Female Total We have to verify 1. YES X Y

Independent versus Dependent Variables: example 2 Consider the example of the 420 employees. Are the variable Smoke (X) independent from the variable College Graduate (Y)? College Graduate Not a College Graduate Total Smoker Nonsmoker Total

Independent versus Dependent Variables: example 2 College Graduate Not a College Graduate Total Smoker Nonsmoker Total We have to verify No independence!!

Independent versus Dependent Variables Two variables are maximally dependent if the contingency table is y 1 … y k … y q y 1 … y k … y q x1x1……xhxh……xpxpx1x1……xhxh……xpxp…… n … 0 0 … 0 … n hq 0 … 0 … n hq 0 … n pk … 0 0 … n pk … 0 n 11 n hq n pk n 11 … n pk … n hq n 11 … n pk … n hqn There is a one-to-one relation between the categories of the two variables

Chi square How caw we measure the “degree” of dependence between two variables? independent Remind that two variables are independent if From these relations we get: n hk * theoretical expected frequencyE n hk * is called theoretical or expected frequency ( E ) since it expresses the frequency of the category h of X and k of Y in condition of independence.

Chi square The observed frequencies n ik are indicated with (O). If the observed frequencies (O) are equal to the expected frequencies (E ) the variables are independent. Chi square. We can build an indicator of independence/dependence between the two variables called Chi square. The formula is It is evident the if Chi square is equal to 0 (O=E ) the two variables are independent.

Chi square: example 1 gender opinion Violence and lack of discipline have become major problems in schools in the United States. A random sample of 300 adults was selected, and they were asked if they favor giving more freedom to schoolteachers to punish students for violence and lack of discipline. The two-way classification of the responses of these adults is represented in the following table. Are the two variables gender and opinion independent? In Favor (F) Against (A) No Opinions (N) Men (M) Women (W)

Chi square: example 1 In Favor (F) Against (A) No Opinion (N) Row Total s Men (M) Women (W) Column Totals In order to compute the chi square we have to compute the expected frequencies as follows:

Chi square: example 1 In Favor (F) Against (A) No Opinion (N) Row Totals Men (M) O 93 (O ) E (105.00) (E ) 70 (59.50) 12 (10.50) 175 Women (W) 87 (75.00) 32 (42.50) 6 (7.50) 125 Column Totals For example

Chi square: example 1 The value of the chi square is different from 0 and hence we should conclude that the two variables are independent.

Chi square: critical value However it can happen that even if the chi square is different from 0, its value is sufficiently small to think that there is independence between the variables of interest. critical value But which value of the chi square can be considered a critical value so that values under this critical value indicate independence and values over this critical value indicate dependence between the two variables? It does not exist a fixed critical value. It is determined time by time depending on the data we are examining by using the methods and the principles of the statistical inference

Chi square: critical value critical value We do not deal with the computation of the critical value. critical value However the critical value is computed from all the Statistical software, included Excel.Rule critical value > chi square independent 1.If the critical value > chi square the two variables can be considered independent critical value < chi square dependent 2.If the critical value < chi square the two variables can be considered dependent in the sense that they influence reciprocally. In the previous example the critical value is It is greater than the value of the chi square (8.252) than we can say that the two variables are independent, that is the opinion of the selected people is not influenced by the gender.

Chi square: example 2  A researcher wanted to study the relationship between gender and owning cell phones. She took a sample of 2000 adults and obtained the information given in the following table. Own Cell PhonesDo Not Own Cell Phones Men Women Looking at the table can we conclude that gender and owning cell phones are related for all adults?

Chi square: example 2 Own Cell Phones (Y) Do Not Own Cell Phones (N) Row Totals Men (M) 640 (588.60) 450 (501.40) 1090 Women (W) 440 (491.40) 470 (418.60) 910 Column Totals We have to compute the expected frequencies

Chi square: example 2 Critical value= The critical value is less than the chi square and hence we can conclude the two variables are dependent, that is owning cell phone depends on gender.

LINEAR REGRESSION LINEAR REGRESSION

LINEAR REGRESSION So far we investigated the relation of independence/dependence between two variables (qualitative or quantitative). However this kind of relation is reciprocal, in the sense that we don’t know if one variable influences the other or vice versa and we don’t know how strong is this relation. Linear regression. If we would like to know if one variable influences the other and how strong this relation is we have to refer to Linear regression. By using the regression analysis we can evaluate the magnitude of change in one variable due to a certain change in another variable and we can predict the value of one variable for a given value of the other variable. (Linear) regression quantitative (Linear) regression is a statistical analysis that evaluates if exists a linear relationship between two quantitative variables, X and Y.

SIMPLE LINEAR REGRESSION Definition simple regression modelindependent dependent. A regression model is a mathematical equation that describes the relationship between two or more variables. A simple regression model includes only two variables: one independent and one dependent. The dependent variable is the one being explained, and the independent variable is the one used to explain the variation in the dependent variable.

Why is it called “regression model” or “regression analysis”? The method was first used to examine the relationship between the heights of fathers and sons. The two were related, of course. But they found that a tall father tended to have sons shorter than himself; a short father tended to have sons taller than himself. The height of sons regressed to the mean. The term "regression" is now used for many sorts of curve fitting. SIMPLE LINEAR REGRESSION linear regression model. A (simple) regression model that gives a straight-line relationship between two variables is called a linear regression model.

LINEAR REGRESSION : example 1 We want to investigate the relation between Incomes (in hundreds of dollars) (X) and Food Expenditures of Seven Households (Y). That is we want to investigate if Income influences Household’s decision about Food Expenditure and how strong is this influence. Income (X)Food Expenditure (Y)

LINEAR REGRESSION : example 1 Scatter plot. We can represent the data with a Scatter plot. A scatter plot is a plot of the values of Y versus the corresponding values of X: Income Food expenditure First household Seventh household

LINEAR REGRESSION : example 1 The scatter plot seems to reveal a linear relationship between the two variables: a linear regression model might be indicated. In the Figure the points (observations) are replaced by a linear model (a) and non linear model (b). Linear Income Nonlinear Income Food Expenditure

LINEAR REGRESSION: the equation How can we write the linear model mathematically? y = a + b x intercept Constant term or y- intercept Slope Dependent variable Independent variable

LINEAR REGRESSION: intercept a ? How can we represent a graphically? The intercept is the Y value of the line when X equals zero. The intercept determines the position of the line on the Y axis. a1a1 a2a2 a3a3 a4a4 Y X 0

LINEAR REGRESSION: slope b ? How can we represent b graphically? The slope quantifies the steepness of the line. It equals the change in Y for each unit change in X. If the slope is positive, Y increases as X increases. If the slope is negative, Y decreases as X increases. X Yb>0 X Yb<0 X Y b1b1 b2b2 b2>b1b2>b1b2>b1b2>b1

LINEAR REGRESSION best” Coming back to the example, among all the possible lines that can interpolate the points in the scatter plot which is the “best” ? Income Food expenditure Choosing the best line (or the line that best describes the relation between X and Y) means finding the “best” a and the “best” b