Correlation Anolysis D Correlation Is a measure of the relationship between two [2] variables. D It Is a measure of how things are related. D The study of how variables are correlated ls called correlation analysis. The following arecomrnon applications of correlation analysis In business and social sciences: D The marketing manager wants to know If price reduction has any Impact on Increasing sales. D The production department wants to know if the number of defective Items produced has anything to do with the machine's age. D The HR department wants to know If its workers' productivity decreases with the number of hours, they put in. D A social activist wants to know if Increasing female literacy has Impacts in Increasing the age of marriage of the girl child. Correlation Analysis High and Low Correlation D High correlation describes a stronger correlation between two variables, wherein a change in the first has a close association with a change in the second. Low correlation describes a weaker correlation, meaning that the two variables are probably not related. Positive, Negative, and No Correlation D Positive oorrelation means that this linear relationship is positive, and the two variables increase or decrease in the same direction. D negative correlation is just the opposite, wherein the relationship line has a negative slope, and the variables change in opposite directions (Le, one variable decreases while the other increases]. D hio oorrelation simply means that the variables behave very differently and thus. have no linear relationship. Correlation Analysis As the corresponding graphs show, we can conclude the following correlations: temperature and ice cream sales: the hotter the day, the higher the ice cream sales. This is a positive correlation. length of workout and body mass index (BMI): the longer the workout, the lower the BMI. This is a negative correlation. shoe size and hair color: show size has no relation to hair color. This has no correlation. Time Spend Emissing vs. BM Positive, Negative, and No Correlation: Graphical Representations. When r is greater than zero, the correlation is positive. When is less than zero, the correlation is negative. When r = 0, there is no correlation. Correlation Coefficient The correlation coefficient gives a mathematical value (r-value) for measuring the strength of the relationship between two (2) variables. Correlation coefficients have a value between -1 to 1. A "0" represents no linear relationship between the two (2) variables at all. A "1" means a perfect positive linear relationship (as one variable increases, the other variable also increases). And "-1" represents an inverse relationship or perfect negative linear relationship (as one variable increases, the other variable decreases). The following are the illustrative representation of correlation coefficients and its interpretations. Scatter Plots & Correlation Examples Perfect Highly Low No Highly Positive Positive Positive Correlation Negative Negative Negative Correlation Correlation Correlation Correlation Correlation Correlation r =1 r =0.8 =0.3 r=0 =-0.3 =-0.8 r =-18 Correlation Coefficient Correlation Interpretation Coefficient 1 Perfect Positive Correlation 0.99 to 0.71 High/strong Positive Correlation 0.70 to 0.31 Moderate Positive Correlation 0.30 to 0.01 Low/weak Positive Correlation 0 No Correlation -0.01 to -0.30 Low/weak Negative Correlation -0.31 to -0.70 Moderate Negative Correlation -0.71 to -0.99 High/strong Negative Correlation -1 Perfect Negative Correlation Correlation Coefficient The height and weight of ten (10) Name Weight (kg) Waist (cm) students were recorded and plotted in Albert 87 101 a graph. If the correlation coefficient is Beth 65 71 0.673 and the graph are as shown. How Cindy 52 62 will you describe the correlation 94 between their heights and weights? David 113 Emily 87 88 Frank 79 87 The correlation coefficient of Gary 59 71 weight and waist size of ten (10) Helen 64 83 friends was calculated and obtained Ida 45 Jeremy 77 85 a value of 0.9438. How will you describe the correlation between their weights and waist sizes?10 Pearson's Product Moment Correlation The Pearson's Product Moment Correlation coefficient (Pearson's r) is a measure of the strength of a linear association between two (2) continuous variables and is denoted by r. Pearson's product-moment correlation coefficient can be used if: The variables can be measured at the interval or ratio level (continuous); o There are no significant outliers; The variables should be approximately normally distributed. Pearson's Product Moment Correlation Hypothesis Test with Pearson's Correlation Pearson's correlation coefficient (r) tells us about the strength of the linear relationship between the two (2) continuous variables. The correlation coefficient is tested to determine whether the linear relationship in the sample data effectively models the relationship in the population. The following are the steps in performing a hypothesis test. 1. State the hypotheses and set the level of significance or alpha level. The null and alternative hypotheses can be written as follow: Ho: There is no significant linear relationship between the two (2) continuous variables. Ha: There is a significant linear relationship between the two (2) continuous variables.12 Pearson's Product Moment Correlation Pearson's Product Moment Correlation Respondents Knowledge Calcium Intake Score(out of 50) (mg/day) 10 450 1050 . A dieticians would want to look at the 42 38 900 relationship between calcium intake 15 525 22 710 and knowledge about calcium in 32 854 sports science students. The table 40 800 14 493 shows the data collected. Is there a 730 relationship between calcium intake 894 940 and knowledge about calcium in sports science students? 1085 13 Pearson's Product Moment Correlation 1. State the hypotheses and set the level of significance. Ho: There is no significant relationship between the calcium intake and knowledge about calcium in sports science students. Ha: There is a significant relationship between the calcium intake and knowledge about calcium in sports science students. a = 0.0514 Pearson's Product Moment Correlation 2. Calculate the degree of freedom and the critical value. To calculate the degree of freedom, use n as the sample size and apply the following formula: df = n-2 The critical value requires the value of the alpha level (a) and the degree of freedom. It can be found using Pearson's correlation table. (Refer to page 1-2 of Pearson's and Spearman's Correlation Table) negative. Note: The critical value is positive if the correlation coefficient r is positive; otherwise, 15 Pearson's Product Moment Correlation 2. Calculate the degree of freedom and critical value. Alpha level df = n - 2 = 20 - 2 = 18 0.2 0.1 0.05 0.02 0.01 0.001 0.951057 0.987688 0.996917 0.999507 0.999877 0.999999 2 0.800000 0.900000 0.950000 0.980000 0.990000 0.999000 3 0.687049 0.805384 0.878389 0.934838 0.958735 0.991139 0.60 0.729299 0.811401 0.917200 5 0.550865 0.669489 0 0.754492 0.832874 0.874526 0.950883 6 0.506727 0.621489 0.706734 0.785720 0.834842 0.924904 0.471589 0.582206 0.666884 0.749776 0.7976 0.898260 0.442796 0.549357 0.631897 0.715459 0.764592 0.872115 0.418662 0.521404 0.602069 0.685095 0.734786 0.847047 10 0.898062 0.497265 0.575983 0.658070 0.707858 0.828305 0.380216 0.476156 0 0.552943 0.633863 0.683528 0.800962 12 0.364562 0.457500 0.532413 0.612047 0.661876 0.779998 8 0.440861 0.518977 0.592270 70 0.641145 0.760851 Critical value 14 0.338282 0 20.425902 0.497309 9 0.574245 0.622591 0.7419% 15 0.327101 0.412360 0.482146 0.557737 0.605506 4./24657 16 0.316958 0.400027 0.468277 0.542548 0.209714 0.708429 17 0.307702 0.388733 0.455531 0 595947 0.575067 0.693163 18 0.299210 0.878341 0.443763 0.515505 0.561435 0.678781 19 0.291384 0.368737 0.432858 0.508397 0.548711 0.66520816 Pearson's Product Moment Correlation 3. Calculate the test statistic (r-value). Given X and Y variables, Pearson's correlation coefficient (r-value) can be computed using the formula: nExy - Ex Zy r= . In Ex -(Ex)Zyz -(Zy)31 where * are the observations in X variable, y are the observations in Y variable, x2 are the squares of x observations, yz are the squares of y observations, xy are the products of x and y observations, and n is the sample size. 17 Pearson's Product Moment Correlation Respondents Knowledge Calcium Intake Score (x) XV V2 3. Calculate the test statistic (r value). (y) 450 4500 101 202500 42 1050 14100 1764 1102500 nExy - ExZy 38 900 34200 1444 810001 15 525 7875 225 275625 VIn Ex - (Xx)?][nEy2 - (Ly)2] 22 710 15620 484 504100 32 854 27328 1024 729316 40 800 32000 1600 64000 14 493 6902 196 243045 20(497143) - (592) (15702) 26 730 18980 676 532900 [20(19852) - (592)2][20(12905468) - (15702)2] 10 32 894 28608 1024 799230 38 940 35720 1441 883600 25 733 18325 625 53728 9942860 - 9295584 13 48 985 47280 2304 970225 28 763 21364 784 582165 (46576)(11556556) 22 583 12826 484 339889 16 45 850 38250 2025 722501 18 798 14364 32 636804 = 0. 882255 18 24 754 18096 576 568510 19 30 805 24150 900 648025 20 43 1085 46655 1849 1177225 sum 592 15702 497143 19852 1290546818 Pearson's Product Moment Correlation 4. Make a statistical decision. - Decision rule using the critical value Suppose r is not between the positive and negative critical values (r is greater than the positive critical value or less than the negative critical value). In that case, the correlation coefficient is significant. Therefore, reject the null hypothesis. On the other hand, if r is between the positive and negative critical value (r is less than the positive critical value or greater than the negative critical value), then the correlation coefficient is not significant. Therefore, we fail to reject the null hypothesis. Decision rule using the p-value Suppose the computed p-value is less than or equal to the set significance level. In that case, the decision is to "Reject the null hypothesis (H.)". On the other hand, if the p-value is greater than the set significance level, the decision is "Fail to reject the null hypothesis (H.)". 5. Conclude. 19 Pearson's Product Moment Correlation . Make a statistical decision. Calcium Intake Pearson's Critical Remarks Description df Decision Knowledge value about High/stron Calcium Reject 0.882255 g positive 18 0.443763 Ho Significant correlation20 Pearson's Product Moment Correlation 5. Conclude. . There is sufficient evidence to conclude that there is significant positive relationship between the calcium intake and knowledge about calcium in sports science students. 21 Pearson's Product Moment Correlation Are there guidelines to interpreting Pearson's correlation coefficient? Yes, the following guidelines have been proposed: Coefficient, r Strength of Positive Negative Association Small .1 to .3 -0.1 to -0.3 Medium 3 to . 5 -0.3 to -0.5 Large .5 to 1.0 -0.5 to -1.022 Pearson's Product Moment Correlation Pearson's r in MS Excel To calculate the test statistic (r-value or p-value) in MS excel, input your data set in a worksheet. In a specific cell, type -PEARSON(B2:B21, Respondents Knowledge Calcium intake C2:C21), where B2:B21 is Scorejout of 50) (me/day) the set of observations on 450 "PEARSON(12 821,CZ CZ1) the X variable and C2:021 PEARSON(amay1, array2) is the set of observations on the Y variable. Then, press Enter. : If we wish to calculate the p-value, first is to find the t-value using the formula shown: Next, calculate the p- value by typing -TDIST(x, deg_freedom,2 ) where x is the t-value, deg_freedom is the degree of freedom, and 2 represents the two-tailed distribution. 23 Spearman's Rank Correlation Spearman's rank correlation evaluates the monotonic relationship between two continuous or ordinal variables. Spearman's rank correlation coefficient measures the strength and direction of the association between two ordinal or continuous variables. Unlike Pearson's r, it is based on the rank values for each variable rather than the raw data. It is denoted by rs, also signified by p (rho). Spearman's correlation coefficient can be used if: . The data is not normal; . The relationship between data is non-linear; . Ordinal variables are being used; and . There are significant outliers.24 Spearman's Rank Correlation Hypothesis Test with Spearman's Rank Correlation Spearman's rank correlation analysis's common desire is to test the null hypothesis that the variables do not have a rank-order relationship in the population represented by the sample. The following are the steps in performing the hypothesis test. 1. State the hypotheses and set the level of significance or alpha level. The null and alternative hypotheses can be written as follow: Ho: There is no significant rank-order relationship between the two (2) continuous variables. Ha: There is a significant rank-order relationship between the two (2) continuous variables. 2. Calculate/identify the critical value. The critical value requires the value of the alpha level (a) and the sample size (n). It can be found using Spearman's correlation table. (Refer to page 3-4 of Pearson's and Spearman's Correlation Table). Note: The critical value is positive if the correlation coefficient r is positive; otherwise, negative. 25 Spearman's Rank Correlation Number of Fish | Rate of Fish A researcher hypothesized that the Store Displayed (X) Quality (Y) store with fewer displayed number 32 6 of fish have healthier fish. He then 2 41 observed and rate the quality of fish 31 W W UT on a 1-10 scale. The table shows the 4 38 data collected. 5 21 7 6 13 9 7 17 9 8 22 8 9 24 6 10 11 11 17 12 20 8a: = 0.05 Spearmon's Rank Correlation _ ' Ho: There is no signicant rank-order relationship between the number of displayed'fish and its quality. ' ' Ha: There is a significant rank-order relationship between the _ number of displayed fish and its quality. Spearmon's Rank Correlation 2. Calculate/identify the critical value. I Alpha level n=12 ennui In on M: III! on: no: um um: ah! I:l 0:0: m: ohm ohm aim aims - Crltical value I . | I I . It 1 Dim uh" 01m . u l mu mm mm mm mm mm mm mm 28 Spearman's Rank Correlation 3. Calculate the test statistic (rs value). Before calculating the rs value, make sure to rank the observations on each variable. (Note: Get the average rank of observations with the same value.) Given X and Y variables, the Spearman's correlation coefficient (rs value) can be computed using the formula: 6 Edz T's = 1 - where n is the sample size, and d are the differences between the ranks of the two variables, and d2 are the squares of differences of the ranks. 29 Spearman's Rank Correlation 3. Calculate the test statistic Number of (rs value) Fish Rate of Fish Store Displayed (X) Quality (Y) Rank (X) Rank (Y) d d2 6 Ed2 32 10 4.5 5.5 30.25 T's = 1 - n3 - n 41 12 3 9 81 31 3 9 1.5 7.5 56.25 38 3 11 1.5 9.5 90.25 5 21 7 6 6.5 -0.5 0.25 6(531) 6 13 9 2 11 81 = 1 (12)3-12 17 9 3.5 11 -7.5 56.25 8 22 8 8.5 -1.5 2.25 9 24 6 8 4.5 3.5 12.25 = -0.857 10 11 11 -10 100 11 17 3.5 6.5 -3 12 20 8 8.5 -3.5 12.25 Sum 531Spearmcin's Rank Correlation 4. Make a statistical decision. . Using the critical value - If the absolute value of the obtained rs is greater than the critical value, then reject the null hypothesis and conclude that there is a rank-order relationship between the two variables. - If the absolute value of the obtained rs is less than the critical value, then the decision Is failed to reject the null hypothesis and conclude that there is no rank-order relationship between the two variables. 5. Conclude. Spearmon's Rank Correlation 4. Make a statistical decision. I Number of fish Displayed I-I-nm _ - - value Qualityof . High/strong . the Fish ' negative 12 . ' ' Significant correlation ' ' ' 0 Since the absolute value of the obtained I\"s is greater than the critical value, then we reject the null hypothesis" 32 Spearman's Rank Correlation 5. Conclude. . There is sufficient evidence to conclude that there is significant negative relationship between the number of displayed fish and its quality. . Stores with fewer displayed fish have healthier fish than those stores with more displayed fish. Spearman's Rank Correlation in MS Excel QUOTIENT . 1 6 1-((6'614)/(12-3)-(32)1) : To calculate the test 30.25 statistic in MS excel, input your data set in a worksheet. : On the next empty column, calculate the rank On the F and G columns. of the observations on the calculate the difference of X variable by typing the rankings and its 12.25 =RANK.AVG (B2, $B$2:$B squares by typing = D2-E2 and =F2^2, respectively. $13,1) where B2 is the first D2 and E2 are the observation in X variable, rankings, and F2 is the 12.25 $B$2:$8$13 is the set of Sum 511 difference 6 Ed2 observations in X variable, Get the sum of the values T's = 1 r s value -1-1 6-6141/1(3249 -(127 and 1 represents the order in column G, then calculate n3 - n 10 for decreasing order the Spearman's correlation and 1 for increasing order). coefficient using the Drag down the cell until all formula shown observations in the X variable get its rank. Do the same for the Y variable.PROBLEM #1 (10 PTS) Students Height| Weight 181 89 2 175 The following table gives the heights 54 and weights of 10 students. 3 143 51 4 190 90 5 160 88 Calculate the Pearson's correlation coefficient and identify the strength of 6 182 75 correlation between the students' height 7 167 58 and weight. 8 178 65 150 41 10 175 70 PROBLEM #2 (10 PTS) Calculate the Pearson's correlation coefficient and identify the strength of correlation between the two variables shown in the table below. Respondents Age (X) Glucose Level (Y) 1 42 98 IN 22 66 26 80 41 74 60 82 56 86 PROBLEM #3 (10 PTS) Calculate the Pearson's correlation coefficient and identify the strength of correlation between the two variables shown in the table below. Respondents Height (cm) Test Score (%) 181 57 169 72 150 57 177 30 145 46 174 78 167 86 160 68 186 39 168 88PROBLEM # 4 (10 PTS) Calculate the Spearman's rank correlation coefficient and identify the strength of correlation between the two (2) variables in the table below. Students Math Score Science score 85 75 83 76 74 92 91 88 86 83 84 84 87 86 90 79 78 94 70 90 PROBLEM #5 (10 PTS) Calculate the Spearman's rank correlation coefficient and identify the strength of correlation between the two (2) variables in the table below. Respondents Self-rated Aggression Peer-rated Aggression 2 3 3 4 COU NO 5 - O UI A W N 8 9 10