Bivariate Analysis Bivariate analysis examines how two variables are related to each other. The most common bivariate statistic is the bivariate correlation (often, simply called “correlation”), which is a number between -1 and +1 denoting the strength of the relationship between two variables. Let’s say that we wish to study how age is related to self-esteem in a sample of 20 respondents, i.e., as age increases, does self-esteem increase, decrease, or remains unchanged. If self-esteem increases, then we have a positive correlation between the two variables, if selfesteem decreases, we have a negative correlation, and if it remains the same, we have a zero correlation. To calculate the value of this correlation, consider the hypothetical dataset shown in Table 14.1. Q u a n t i t a t i v e A n a l y s i s : D e s c r i p t i v e S t a t i s t i c s | 123 Figure 14.2. Normal distribution Table 14.1. Hypothetical data on age and self-esteem The two variables in this dataset are age (x) and self-esteem (y). Age is a ratio-scale variable, while self-esteem is an average score computed from a multi-item self-esteem scale measured using a 7-point Likert scale, ranging from “strongly disagree” to “strongly agree.” The histogram of each variable is shown on the left side of Figure 14.3. The formula for calculating bivariate correlation is: where rxy is the correlation, x and y are the sample means of x and y, and sx and sy are the standard deviations of x and y. The manually computed value of correlation between age and self-esteem, using the above formula as shown in Table 14.1, is 0.79. This figure indicates 124 | S o c i a l S c i e n c e R e s e a r c h that age has a strong positive correlation with self-esteem, i.e., self-esteem tends to increase with increasing age, and decrease with decreasing age. Such pattern can also be seen from visually comparing the age and self-esteem histograms shown in Figure 14.3, where it appears that the top of the two histograms generally follow each other. Note here that the vertical axes in Figure 14.3 represent actual observation values, and not the frequency of observations (as was in Figure 14.1), and hence, these are not frequency distributions but rather histograms. The bivariate scatter plot in the right panel of Figure 14.3 is essentially a plot of self-esteem on the vertical axis against age on the horizontal axis. This plot roughly resembles an upward sloping line (i.e., positive slope), which is also indicative of a positive correlation. If the two variables were negatively correlated, the scatter plot would slope down (negative slope), implying that an increase in age would be related to a decrease in self-esteem and vice versa. If the two variables were uncorrelated, the scatter plot would approximate a horizontal line (zero slope), implying than an increase in age would have no systematic bearing on self-esteem. Figure 14.3. Histogram and correlation plot of age and self-esteem After computing bivariate correlation, researchers are often interested in knowing whether the correlation is significant (i.e., a real one) or caused by mere chance. Answering such a question would require testing the following hypothesis: H0: r = 0 H1: r ≠ 0 H0 is called the null hypotheses, and H1 is called the alternative hypothesis (sometimes, also represented as Ha). Although they may seem like two hypotheses, H0 and H1 actually represent a single hypothesis since they are direct opposites of each other. We are interested in testing H1 rather than H0. Also note that H1 is a non-directional hypotheses since it does not specify whether r is greater than or less than zero. Directional hypotheses will be specified as H0: r ≤ 0; H1: r > 0 (if we are testing for a positive correlation). Significance testing of directional hypothesis is done using a one-tailed t-test, while that for non-directional hypothesis is done using a two-tailed t-test. Q u a n t i t a t i v e A n a l y s i s : D e s c r i p t i v e S t a t i s t i c s | 125 In statistical testing, the alternative hypothesis cannot be tested directly. Rather, it is tested indirectly by rejecting the null hypotheses with a certain level of probability. Statistical testing is always probabilistic, because we are never sure if our inferences, based on sample data, apply to the population, since our sample never equals the population. The probability that a statistical inference is caused pure chance is called the p-value. The p-value is compared with the significance level (α), which represents the maximum level of risk that we are willing to take that our inference is incorrect. For most statistical analysis, α is set to 0.05. A p-value less than α=0.05 indicates that we have enough statistical evidence to reject the null hypothesis, and thereby, indirectly accept the alternative hypothesis. If p>0.05, then we do not have adequate statistical evidence to reject the null hypothesis or accept the alternative hypothesis