Chi-Square Test

qi- square off Test Chi-square is a statistical test usually utilise to compare find selective information with data we would expect to obtain according to a specific system. For example, if, according to Mendels laws, you pass judgment 10 of 20 offspring from a cross to be male and the material ob parcel outd look was 8 males, then you might want to hunch about the unplayfulness to fit mingled with the observed and judge. Were the remainders (differences among observed and expected) the expiry of chance, or were they cal testing groundle to bleak(prenominal) factors.How much deviation can pass away before you, the investigator, must conclude that something new(prenominal) than chance is at work, ca apply the observed to differ from the expected. The chi-square test is always testing what scientists call the null possible action, which states that in that respect is no significant difference among the expected and observed result. The formula for reason chi-sq uare ( pic2) is pic2= pic(o-e)2/e That is, chi-square is the sum of the squared difference between observed (o) and the expected (e) data (or the deviation, d), divided by the expected data in all workable categories.For example, suppose that a cross between two pea plants yields a population of 880 plants, 639 with kibibyte get outds and 241 with yellow seeds. You are asked to propose the ge nonypes of the parents. Your hypothesis is that the allele for green is dominant to the allele for yellow and that the parent plants were both heterozygous for this trait. If your hypothesis is true, then the predicted proportion of offspring from this cross would be 31 (based on Mendels laws) as predicted from the results of the Punnett square (Figure B. ). Figure B. 1 Punnett forthrightly. Predicted offspring from cross between green and yellow-seeded plants. Green (G) is dominant (3/4 green 1/4 yellow). To nume yard pic2 , first turn back the number expected in apiece category. If the ratio is 31 and the total number of observed individuals is 880, then the expected numeric assesss should be 660 green and 220 yellow. pic Chi-square requires that you use quantitative observes, not percentages or ratios. pic Then calculate pic2 using this formula, as shown in Table B. . Note that we describe a value of 2. 668 for pic2. But what does this number mean? Heres how to interpret the pic2 value 1. modulate degrees of freedom (df). Degrees of freedom can be compute as the number of categories in the problem negative 1. In our example, there are two categories (green and yellow) therefore, there is I degree of freedom. 2. Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis. The relative standard commonly used in biological research is p 0. 05.The p value is the opportunity that the deviation of the observed from that expected is due to chance wholly (no other forces acting). In this case, using p 0. 05, you would expe ct any deviation to be due to chance alone 5% of the time or less. 3. Refer to a chi-square distribution carry over (Table B. 2). Using the appropriate degrees of freedom, locate the value impending to your careful chi-square in the table. Determine the closestp ( prospect) value associated with your chi-square and degrees of freedom. In this case (pic2=2. 68), the p value is about 0. 10, which path that there is a 10% probability that any deviation from expected results is due to chance only. Based on our standard p 0. 05, this is within the course of acceptable deviation. In terms of your hypothesis for this example, the observed chi-squareis not significantly different from expected. The observed numbers are consistent with those expected under Mendels law. Step-by-Step Procedure for Testing Your Hypothesis and Calculating Chi-Square 1. State the hypothesis being tested and the predicted results.Gather the data by conducting the proper experiment (or, if work genetics pro blems, use the data provided in the problem). 2. Determine the expected numbers for each observational class. Remember to use numbers, not percentages. pic Chi-square should not be cypher if the expected value in any category is less than 5. pic 3. Calculate pic2 using the formula. Complete all calculations to three significant digits. Round off your answer to two significant digits. 4. Use the chi-square distribution table to condition importation of the value. . Determine degrees of freedom and locate the value in the appropriate column. b. Locate the value closest to your calculated pic2 on that degrees of freedom df row. c. Move up the column to determine the p value. 5. State your conclusion in terms of your hypothesis. a. If the p value for the calculated pic2 is p 0. 05, accept your hypothesis. The deviation is small enough that chance alone accounts for it. A p value of 0. 6, for example, means that there is a 60% probability that any deviation from expected is due to ch ance only.This is within the range of acceptable deviation. b. If the p value for the calculated pic2 is p 0. 05, reject your hypothesis, and conclude that some factor other than chance is operating for the deviation to be so great. For example, a p value of 0. 01 means that there is only a 1% chance that this deviation is due to chance alone. Therefore, other factors must be involved. The chi-square test will be used to test for the goodness to fit between observed and expected data from several laboratory investigations in this lab manual. Table B. 1 Calculating Chi-Square Green Yellow Observed (o) 639 241 Expected (e) 660 220 refraction (o e) -21 21 Deviation2 (d2) 441 441 d2/e 0. 68 2 pic2 = picd2/e = 2. 668 . . Table B. 2 Chi-Square Distribution Degrees of Freedom Probability (p) (df) 0. 95 0. 90 Source R. A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed. , Table IV, Oliver & Boyd, Ltd. , Edinburgh, by per mit of the authors and publishers.Main Page Introduction and Objectives Scientific Investigation Experimental Procedures Writing Procedures Mendelian Inheritance Monohybrid and Dihybrid Exercises Reference Mis carrelaneous Scientific Writing Chi-Square Test Graphing Techniques Chi-Square Test Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendels laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the goodness to fit between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors.How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-square test is alwa ys testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result. The formula for calculating chi-square ( pic2) is pic2= pic(o-e)2/e That is, chi-square is the sum of the squared difference between observed (o) and the expected (e) data (or the deviation, d), divided by the expected data in all possible categories. For example, suppose that a cross between two pea plants yields a population of 880 plants, 639 with green seeds and 241 with yellow seeds. You are asked to propose the geno references of the parents.Your hypothesis is that the allele for green is dominant to the allele for yellow and that the parent plants were both heterozygous for this trait. If your hypothesis is true, then the predicted ratio of offspring from this cross would be 31 (based on Mendels laws) as predicted from the results of the Punnett square (Figure B. 1). Figure B. 1 Punnett Square. Predicted offspring from cross betw een green and yellow-seeded plants. Green (G) is dominant (3/4 green 1/4 yellow). To calculate pic2 , first determine the number expected in each category. If the ratio is 31 and the total number of observed individuals is 880, then the expected numerical values should be 660 green and 220 yellow. picChi-square requires that you use numerical values, not percentages or ratios. pic Then calculate pic2 using this formula, as shown in Table B. 1. Note that we get a value of 2. 668 for pic2. But what does this number mean? Heres how to interpret the pic2 value 1. Determine degrees of freedom (df). Degrees of freedom can be calculated as the number of categories in the problem minus 1. In our example, there are two categories (green and yellow) therefore, there is I degree of freedom. 2. Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis. The relative standard commonly used in biological research is p 0. 05.The p value is the probability that t he deviation of the observed from that expected is due to chance alone (no other forces acting). In this case, using p 0. 05, you would expect any deviation to be due to chance alone 5% of the time or less. 3. Refer to a chi-square distribution table (Table B. 2). Using the appropriate degrees of freedom, locate the value closest to your calculated chi-square in the table. Determine the closestp (probability) value associated with your chi-square and degrees of freedom. In this case (pic2=2. 668), the p value is about 0. 10, which means that there is a 10% probability that any deviation from expected results is due to chance only. Based on our standard p 0. 05, this is within the range of acceptable deviation.In terms of your hypothesis for this example, the observed chi-squareis not significantly different from expected. The observed numbers are consistent with those expected under Mendels law. Step-by-Step Procedure for Testing Your Hypothesis and Calculating Chi-Square 1. State the hypothesis being tested and the predicted results. Gather the data by conducting the proper experiment (or, if working genetics problems, use the data provided in the problem). 2. Determine the expected numbers for each observational class. Remember to use numbers, not percentages. pic Chi-square should not be calculated if the expected value in any category is less than 5. pic 3.Calculate pic2 using the formula. Complete all calculations to three significant digits. Round off your answer to two significant digits. 4. Use the chi-square distribution table to determine significance of the value. a. Determine degrees of freedom and locate the value in the appropriate column. b. Locate the value closest to your calculated pic2 on that degrees of freedom df row. c. Move up the column to determine the p value. 5. State your conclusion in terms of your hypothesis. a. If the p value for the calculated pic2 is p 0. 05, accept your hypothesis. The deviation is small enough that chance alone accounts for it. A p value of 0. , for example, means that there is a 60% probability that any deviation from expected is due to chance only. This is within the range of acceptable deviation. b. If the p value for the calculated pic2 is p 0. 05, reject your hypothesis, and conclude that some factor other than chance is operating for the deviation to be so great. For example, a p value of 0. 01 means that there is only a 1% chance that this deviation is due to chance alone. Therefore, other factors must be involved. The chi-square test will be used to test for the goodness to fit between observed and expected data from several laboratory investigations in this lab manual. Table B. 1 Calculating Chi-Square Green Yellow Observed (o) 639 241 Expected (e) 660 220 Deviation (o e) -21 21 Deviation2 (d2) 441 441 d2/e 0. 68 2 pic2 = picd2/e = 2. 668 . . Table B. 2 Chi-Square Distribution Degrees of Freedom Probability (p) (df) 0. 95 0. 90 Frequency DistributionsOne important place of statistical tests allows us to test for deviations of observed frequencies from expected frequencies. To introduce these tests, we will start with a simple, non-biological example. We want to determine if a coin is fair. In other words, are the odds of flipping the coin heads-up the same as tails-up. We collect data by flipping the coin 200 generation. The coin landed heads-up 108 times and tails-up 92 times. At first glance, we might suspect that the coin is biased because heads resulted more often than than tails. However, we have a more quantitative way to die our results, a chi-squared test. To perform a chi-square test (or any other statistical test), we first must establish our null hypothesis.In this example, our null hypothesis is that the coin should be equally likely to land head-up or tails-up every time. The null hypothesis allows us to state expected frequencies. For 200 tosses, we would expect degree centigrade heads and century tails. The succ eeding(a) step is to prepare a table as follows. Heads Tails correspond Observed 108 92 200 Expected 100 100 200 Total 208 192 400 The Observed values are those we gather ourselves. The expected values are the frequencies expected, based on our null hypothesis. We total the rows and columns as indicated. Its a good idea to make sure that the row totals equal the column totals (both total to 400 in this example). Using probability theory, statisticians have devised a way to determine if a frequency distribution differs from the expected distribution. To use this chi-square test, we first have to calculate chi-squared. Chi-squared = ? (observed-expected)2/(expected) We have two classes to consider in this example, heads and tails. Chi-squared = (100-108)2/100 + (100-92)2/100 = (-8)2/100 + (8)2/100 = 0. 4 + 0. 64 = 1. 28 Now we have to consult a table of critical values of the chi-squared distribution. Here is a portion of such a table. df/prob. Types of Data There are basically two types of random variables and they yield two types of data numerical and categorical. A chi square (X2) statistic is used to investigate whether distributions of categorical variables differ from one another. Basically categorical variable yield data in the categories and numerical variables yield data in numerical form. Responses to such questions as What is your major? or Do you own a car? are categorical because they yield data such as biological science or no. In contrast, responses to such questions as How tall are you? or What is your G. P. A.? are numerical. Numerical data can be either distinct or continuous. The table below may help you see the differences between these two variables. Data Type Question Type Possible Responses Categorical What is your wake? male or female Numerical Disrete- How many cars do you own? two or three Numerical Continuous How tall are you? 72 inches Notice that discrete data arise fom a counting process, while continuous data aris e from a measuring process.The Chi Square statistic compares the tallies or counts of categorical responses between two (or more) independent groups. (note Chi square tests can only be used on actual numbers and not on percentages, proportions, means, etc. ) 2 x 2 Contingency Table There are several types of chi square tests depending on the way the data was amass and the hypothesis being tested. Well begin with the simplest case a 2 x 2 contingency table. If we set the 2 x 2 table to the full general notation shown below in Table 1, using the letters a, b, c, and d to denote the contents of the cells, then we would have the following table Table 1. command notation for a 2 x 2 contingency table. Variable 1 Variable 2 Data type 1 Data type 2 Totals Category 1 a b a + b Category 2 c d c + d Total a + c b + d a + b + c + d = N For a 2 x 2 contingency table the Chi Square statistic is calculated by the formula pic Note notice that the four components of the denominator are the f our totals from the table columns and rows. Suppose you conducted a dose trial on a group of animals and you hypothesized that the animals receiving the drug would show increased heart rates compared to those that did not receive the drug.You conduct the study and collect the following data Ho The proportion of animals whose heart rate increased is independent of drug treatment. Ha The proportion of animals whose heart rate increased is associated with drug treatment. Table 2. Hypothetical drug trial results. HeartRate NoHeartRate Total Increased Increase Treated 36 14 50 Not do by 30 25 55 Total 66 39 105 Applying the formula above we getChi square = 105(36)(25) (14)(30)2 / (50)(55)(39)(66) = 3. 418 Before we can arise we eed to know how many degrees of freedom we have. When a comparison is made between one sample and another, a simple rule is that the degrees of freedom equal (number of columns minus one) x (number of rows minus one) not counting the totals for rows or columns. For our data this gives (2-1) x (2-1) = 1. We now have our chi square statistic (x2 = 3. 418), our predetermined alpha take of significance (0. 05), and our degrees of freedom (df=1). Entering the Chi square distribution table with 1 degree of freedom and reading along the row we find our value of x2 (3. 418) lies between 2. 706 and 3. 841.The corresponding probability is between the 0. 10 and 0. 05 probability levels. That means that the p-value is above 0. 05 (it is actually 0. 065). Since a p-value of 0. 65 is greater than the conventionally accepted significance level of 0. 05 (i. e. p0. 05) we fail to reject the null hypothesis. In other words, there is no statistically significant difference in the proportion of animals whose heart rate increased. What would happen if the number of control animals whose heart rate increased dropped to 29 instead of 30 and, consequently, the number of controls whose hear rate did not increase changed from 25 to 26? Try it. Notice tha t the new x2 value is 4. 25 and this value exceeds the table value of 3. 841 (at 1 degree of freedom and an alpha level of 0. 05). This means that p 0. 05 (it is now0. 04) and we reject the null hypothesis in favor of the alternative hypothesis the heart rate of animals is different between the treatment groups. When p 0. 05 we primarily refer to this as a significant difference. Table 3. Chi Square distribution table. probability level (alpha) Df 0. 5 0. 10 0. 05 A 10 42 52 a 33 15 48 Totals 43 57 100 The penotypic ratio 85 of the A type and 15 of the a-type (homozygous recessive). In a monohybrid cross between two heterozygotes, however, we would have predicted a 31 ratio of phenotypes. In other words, we would have expected to get 75 A-type and 25 a-type. Are or resuls different? pic Calculate the chi square statistic x2 by completing the following steps 1. For each observed number in the table subtract the corresponding expected number (O E). 2. Square the difference (O E)2 . 3. Divide the squares obtained for each cell in the table by the expected number for that cell (O E)2 / E . 4. Sum all the values for (O E)2 / E. This is the chi square statistic. For our example, the calculation would be Observed Expected (O E) (O E)2 (O E)2/ E a-type 15 25 10 100 4. 0 Total 100 100 Suppose you have the following categorical data set. Table . Incidence of three types of malaria in three equatorial regions. Asia Africa South America Totals 14 23. 04 9. 04 81. 72 3. 546 45 36. 00 9. 00 81. 00 2. 5 2 20. 64 18. 64 347. 45 16. 83 5 15. 36 10. 36 107. 33 6. 99 53 24. 00 29. 00 841. 00 35. 04 53 34. 40 18. 60 345. 96 10. 06 45 25. 60 19. 40 376. 36 14. 70 2 40. 00 38. 00 1444. 00 36. 10 Chi Square = 125. 516 Degrees of Freedom = (c 1)(r 1) = 2(2) = 4 Table 3.Chi Square distribution table. probability level (alpha) Df 0. 5 0. 10 0. 05 0. 02 0. 01 0. 001 1 0. 455 2. 706 3. 841 5. 412 6. 635 10. 827 2 1. 386 4. 605 5. 991 7. 824 9. 210 13. 815 3 2. 366 6. 251 7. 815 9. 837 11. 345 16. 268 4 3. 357 7. 779 9. 488 11. 668 13. 277 18. 465 5 4. 351 9. 236 11. 070 13. 388 15. 086 20. 517 rid of Ho because 125. 516 is greater than 9. 488 (for alpha 0. 05) Thus, we would reject the null hypothesis that there is no relationship between location and type of malaria. Our data tell us there is a relationship between type of malaria and location, but thats all it says.Follow the link below to access a umber-based program for calculating Chi Square statistics for contingency tables of up to 9 rows by 9 columns. Enter the number of row and colums in the spaces provided on the page and click the submit button. A new form will appear asking you to enter your actual data into the cells of the contingency table. When finished entering your data, click the calculate now button to see the results of your Chi Square analysis. You may wish to print this last page to keep as a record. Chi Square, This page was created as part of the M athbeans Project. The java applets were created by David Eck and modified by Jim Ryan. The Mathbeans Project is funded by a grant from the National Science Foundation DUE-9950473.