Goodness of Fit and Test for Independence

Transcription

1 Goodness of Fit and Test for Independence In this class we will consider the χ 2 test. This is a hypothesis testing method used to decide if the data correspond to the normal curve. It also can be extended to the problem of testing if two variables are independent. We will consider the following topics: How to test if the data correspond to the model The χ 2 curve A χ 2 test for independence between the rows and the columns of a table

2 The Chi-square test How well does the data fit the model? This is the question that is tackled by the χ 2 test. Consider the next data that correspond to 60 rolls of a die Do they correspond to a loaded die?

3 In statistical terms the question is if the data correspond to 60 samples at random with replacement from the box: If they do, then we should observe, approximately, the same number of occurrences for each one of the six numbers. For previous tests we have seen that the idea is to compute the standardized difference between the observed value and the value that is expected under the null hypothesis.

4 The problem in this case is that we have six possible values, one for each ticket in the box. In fact, if the die is not loaded we expect to see each number 10 times. The values we observe from the samples we have are Value sum Observed Expected To measure the distance between the observed values and the expected values we use the statistics χ 2 = sum of (observed frequency - expected frequency) 2 expected frequency

5 The idea is that, if the observed frequencies are far from the expected frequency, the distance will be large. So, large values of χ 2 are considered as evidence against the hypothesis that the sample corresponds to the model. From the data in the previous table we obtain (4 10) (6 10) (17 10) (16 10) (8 10)2 10 (9 10) = 14.2

6 The χ 2 curve How do we decide if the observed value of χ 2 is large? The χ 2 curve was calculated by a statistician named Pearson to provide the P-values for the χ 2 test. It depends on the degrees of freedom (like the Student s t), so there is one curve for each value of the degrees of freedom DF 10 DF 15 DF 20 DF The P-value for the χ 2 test is the area under the χ 2 curve and to the right of the χ 2 value with the number of Degrees of Freedom calculated as degrees of freedom = number of terms in χ

7 For the die example we have that Degrees of Freedom = 6-1 = 5 So we need to find the area under a χ 2 curve with 5 DF to the right of We either use a calculator or the table at the end of the book. The table is similar to the Student s t table. It has a row for each number of DF and the columns correspond to probabilities. We observe that, for 5 degrees of freedom DF 90% 50% 10% 5% 1% so there is no entry for The closest value is that corresponds to 1% chance. Thus the P-value for this example is slightly larger than 1%.

8 The conclusion is that the box model of a fair die does not correspond to the samples taken. Two issues are important in this example: 1. The model is fully specified since there are no parameters to estimate. 2. The expected frequency in each line of the table that is used to produce the test statistics is large (5 or more).

9 Testing independence The χ 2 -test can be used to test for independence. The goal is to test if the variables that correspond to those criteria are independent. We will illustrate with what is usually called a Contingency table. Example: A sample of 2,237 Americans age 25 to 34 was used to produce the following table Men Women Totals Right-handed 934 1,070 2,004 Left-handed ambidextrous total 1,067 1,170 2,237

10 A better view of the data is obtained when the table is converted to percentages. Men Women Right-handed 87.5% 91.5% Left-handed 10.6% 7.9% ambidextrous 1.9% 0.7% Let s suppose that this data come from sampling at random with replacement from a box with one ticket for each American. There are 6 types of tickets right-handed man left-handed man ambidextrous man right-handed woman left-handed woman ambidextrous woman We have two variables, one is gender and the other is handedness. Under the null hypothesis we will assume that the two variables are independent. That is, the true percentage of right-handed men equals the true percentage of right-handed women and so on. The alternative hypothesis states that the true percentage of right-handed men is different from the true percentage of right-handed women and so on.

11 To make a χ 2 -test we need the expected values corresponding to each cell in the table. How do we get the expected values? The percentage of right-handed in the sample is 2, % 89.6% 2, 237 If handedness and gender are independent then the number of right-handed men and right handed women should be, respectively 89.6% of 1, and 89.6% of 1, 170 1, 048 This can be applied to all other cells in the table. We could have also just computed, 2, 004 1, 067 = 956 2, 237

12 In general the formula for the expected value of a given cell is the calculations for the example yield row total column total total of the table Men Women Right-handed 956 1,048 Left-handed ambidextrous Now we can compute the χ 2 -test using the differences between the original table with the observed values and the table of expected values χ 2 = ( ) (1, 070 1, 048)2 1, (113 98) (92 107) (20 13) (8 15)

13 To decide whether this observed value of the χ 2 -test is significantly large or not, we need to obtain the P-value using the χ 2 curve. We first need to compute the number of DF. We previously computed the DF as the number of data minus 1. So in this case the result would be DF=5. This is WRONG! The reason is that the model is not fully specified since we don t know the true percentages in the box and we had to estimate them. We loose one degree of freedom for each parameter we have to estimate. When testing for independence in a table with m rows and n columns, there are (m 1) (n 1) DF In the example we have m = 3 and n = 2, so we have 2 DF. The area to the right of 12 for a χ 2 curve with 2 DF is less than 1%. So there is significant evidence against the null.

14 Example: A random sample of three types of gastropod shells (hermit crab) was collected, totaling 314 samples. Each shell was classified as being occupied or empty. The goal of the study is to infer if the shell species is independent of whether it is occupied. The data are reported in the following table Species Occupied Empty Total Austrocochlea Bembicium Cirithid Total

15 The table of the expected observations is Then we have that Species Occupied Empty Total Austrocochlea Bembicium Cirithid Total χ 2 = ( ) ( ) ( ) ( )2 ( )2 ( ) = 45.52

16 To calculate the P-value we need to use a χ 2 curve with (3 1) (2 1) = 2 degrees of freedom. From the corresponding table we conclude that the P-value is smaller than 1%. The conclusion is that there is very strong evidence against the null hypothesis, so the species are not independent of the occupation status.

17 Correlation The correlation gives a measure of the linear association between two variables. It is a coefficient that does not depend on the units that are used to measure the data and is bounded between -1 and 1. In this class we will consider the following points: Scatterplots Defining and computing the correlation The effect of changing the variables units The effect of non-linear trends and outliers

18 Correlation How do you study the relationship between two variables? Pearson considered the data corresponding to the heights of 1,078 fathers and their son s at maturity. The plot appears on page 120 of the textbook. A list of these data is difficult to understand, but the relationship between the two variables can be visualized using a scatter diagram, where each pair father-son is represented as a point in a plane. The x-coordinate corresponds to the father s height and the y-coordinate to the son s. We observe that the taller the father the taller son, as a general tendency. This corresponds to a positive association. In this example we consider the height of the father as an independent variable and the height of the son as a dependent variable.

19 Scatter Diagrams Degrees August Temperatures vs Elevation in Northern California feet The figure shows the scatter diagram of the temperature in 58 locations in Northern California on August These data are plotted against the elevations of the station where the measurement was recorded. We can see that there is a tendency for locations at high elevations to have lower temperatures compared to those at low elevations.

20 The correlation coefficient We have seen that mean and standard deviation provide a description of the behavior of a sample for a given variable. When we consider two variables we can calculate the mean and the standard deviation of each of them, but none of those four numbers will give a measure of the association between the two variables. The correlation coefficient gives a measure of the linear association of two variables That is, when the scatter plot of the two variables is very close to the straight line we have a correlation that is close to one.

21 A zero correlation corresponds to a diagram where the data are widely scattered around the line. The correlation coefficient is usually denoted by r and takes values between -1 and 1 A negative coefficient means that the data are clustered around lines with a negative slope. That is, as one variable increases, the other one decreases. The closer r is to 1 the stronger the positive linear association between the variables. The closer r is to -1 the stronger the negative linear association between the variables. When r is equal to 1 or -1 there is total linear association between the variables, this implies that all points lie on a line. The slope of the line is positive for r = 1 and negative for r = 1.

22 Computing the correlation coefficient The procedure to compute the correlation coefficients is the following 1. Convert each variable to standard units 2. Calculate the average of the products The result is the correlation coefficient. The formula is given by r = average of ( x in standard units y in standard units )

23 Example Consider the data x y Each pair of numbers in the table corresponds to a subject. 1. Convert x to standard units. The average of the x-values is 4, the SD is Convert y to standard units. The mean of the y-values is 7 and the SD is Compute the products of the standard units of the x-values and the y-values Take the average of the products r = = 0.40

24 Chitons: Consider the data in the table Length Width Length Width These correspond to the lengths and the widths of 10 chitons. We have that the averages are for the lengths and 5.64 for the widths. The standard deviations are and respectively. Thus the transformation to standard units yields Length Width Length Width Product Then r = is the average of the products. This implies that there is a strong positive linear association between the length and the width of the chitons.

25 Positive correlation y Suppose that this scatterdiagram corresponds to the standard units of the data. The signs of the quadrants are determined by the sign of the products. There are more points in the positive quadrants than in the negative quadrants. So the average is positive yielding a positive correlation x

26 Negative correlation y In this case there are more points in the negative quadrants than in the positive quadrants. So the average is negative, yielding a negative correlation x

27 Features of the correlation coefficient Suppose you measure the correlation between the temperature in New York and that in Boston during a month. Do you expect to get the same correlation if you measure it in Celsius than if you measure it in Fahrenheit? Recall the process of obtaining the correlation. The first step is to convert the samples of both variables to standard units. This implies that the correlation does not depend on units. Thus, no matter if you use Celsius of Fahrenheit, you will get the same correlation. On the other hand, since the correlation depends on the product of the two variables, it does not matter whether you consider the correlation the temperature in NY and that in Boston or the correlation between the temperature in Boston and that in NY.

28 The correlation coefficient has the following properties The correlation is not affected when the two variables are interchanged. The correlation is not changed if the same number is added to all the values of one of the variables. The correlation is not changed if all the values of one of the variables is multiplied by the same positive number. It will change sign if the number is negative.

29 Non-linear association Correlation is useful only when measuring the degree of linear association between two variables. That is, how much the values from two variables cluster around a straight line. y The variables in this plot have an obvious non-linear association. Nevertheless the correlation between them is 0.3. This is because the points are clustered around a curve that has the shape of a parabola and not a straight line. x

30 Examples Find the correlation coefficient for each of the following data sets: Example 1: x y The average of x is 2, the SD is 1. The average of y is 3 and the SD is 2. Then, the standardized values are x y x y and the average of the last column is

31 Example 2: x y x has not changes, y has average equal to 2 and a SD equal to 1. Then, the standardized values are x y x y and the average of the last column is 0.3.

32 Example 3: x y We observe that y = 2 x and so there is total linear association between x and y, implying that the correlation is 1.