Analyzing Quantitative Data

RELATIONS ONE VARIABLE TO THE OTHER

In research projects with multiple variables, we typically want to know how one variable is related to another in addition to the descriptive statistics of the variables. That is, the kind, direction, and significance of the bivariate correlations between the variables utilized in the study—that is, the relationship between any two variables among the variables used in the study—are what we would like to see. There are nonparametric tests available to evaluate the association between variables that are measured using either an ordinal or nominal scale. Two ordinal variables are compared using the Spearman's rank correlation and the Kendall's rank correlation. Relationships between interval and/or ratio variables can be examined using a correlation matrix.

LINKAGE BETWEEN TWO NOMINAL VARIABLE SETS

There are situations when we would like to determine whether two nominal variables are independent of one another or if there is a correlation between them. As instances: Is there a correlation between an individual's decision to buy or not buy a product and whether they watch a television advertisement for it? (2) Is a person's skin tone—white or nonwhite—related to the kind of work they do (white-collar or blue-collar jobs)? Sorting data into groups or categories and looking for statistically significant associations allows for these kinds of comparisons. For instance, based on a frequency count, we may gather information from a sample of 55 people whose skin tone and type of employment might be displayed in a two-by-two contingency table.

It appears that there is a definite trend toward white people working in white-collar jobs. White-collar employment are held by very few non-White people. Therefore, there appears to be a correlation between skin tone and the kind of work performed; the two do not appear to be unrelated. The chi-square (c2) test is a nonparametric test that can be used to statistically confirm whether or not the observed pattern is the result of chance. Nonparametric tests, as we all know, are employed when it is not possible to assume that distributions are normal, as in the case of nominal or ordinal data.

With computer analysis, the c2 statistic and its significance level can be found for any collection of nominal data. Thus, the c2 (chi-square) statistic is useful when examining differences in associations among nominally scaled variables. In the example above, the variables would be the color of skin and the type of employment. The null hypothesis would be made to declare that there is no significant association between the two variables, and the alternate hypothesis would state that there is. The degrees of freedom (df), which indicate whether or not there is a significant link between two nominal variables, are related to the chi-square statistic. One fewer degree of freedom exists than there are cells in the rows and columns. If any four cells (two in a column and two in a row), then the number of degrees of freedom would be 1: [(2 – 1) × (2 – 1)].

For two nominal variables with many levels, the c2 statistic can also be applied. It would be interesting to know, for example, if four employee groups—production, sales, marketing, and R&D staff—react differently to a policy (that is, with no interest at all, mild interest, considerable interest, and extreme interest). Here, the data is cross-tabulated in 16 cells, or classified according to the four employee groups and the four categories of interest, to provide the c2 value for the test of independence. Here, there will be nine degrees of freedom: [(4–1) × (4–1)].

Thus, the c2 test of significance aids in determining the relationship between two nominal variables. In addition to the c2 test, alternative methods for establishing the association between two nominally scaled variables include the Fisher exact probability test and the Cochran Q test.

CORRELATIONS

The direction, intensity, and significance of the bivariate correlations between all the variables that were measured at an interval or ratio level will be shown in a Pearson correlation matrix. By evaluating the changes in one variable while another varies as well, the correlation is determined. Let's assume, for the purpose of simplicity, that we have data on sales and price for two distinct products. For each product, the volume of sales at each price point can be plotted.

Using a method that accounts for the two sets of data—in this case, distinct sales volumes at various prices—one may calculate a correlation coefficient that shows the strength and direction of the association. In theory, two variables might theoretically have a perfect positive correlation of 1.0 (plus 1) or a perfect negative correlation of -1.0 (minus 1). However, when evaluating correlations between any two variables that are supposed to differ from one another, neither of these will actually be detected.

We need to determine whether any correlation between two variables is meaningful, even though it could range from -1.0 to +1.0 (i.e., if it has occurred merely by chance or if there is a high possibility of its actual existence). As is well knowledge, the conventionally recognized threshold for significance in social science research is p = 0.05. This means that there is a mere 5% possibility that the relationship between the two variables is not real, and 95 times out of 100, we may be certain that there is a true or substantial correlation.

We can conclude that there is a positive association between two variables A and B if their correlation coefficient (r = 0.56) is 0.56 (p < 0.01). The probability that this relationship is false is 1% or less. In other words, this correlation should exist more than 99% of the time. Additionally, the 0.56 correlation coefficient shows that the variables account for 31.4% of each other's variation (0.56 square).Although we are unsure of which variable causes which, we do know that the two are related. Therefore, by looking at the correlation between the two variables, a hypothesis that suggests a substantial positive (or negative) relationship between the two of them can be examined.

For interval- and ratio-scaled variables, the Pearson correlation coefficient is adequate; for ordinal-scaled variables, the Spearman Rank or Kendall's tau coefficients are appropriate. By selecting the suitable option, figuring out the variables, and looking for the right parametric or nonparametric statistics, one can find any bivariate connection.

PART 1 OF EXCELSIOR ENTERPRISES: DESCRIPTIVE STATISTICS

The Excelsior Enterprises study's interval-scaled items yielded descriptive statistics like variance, averages, standard deviations, and maximum and minimum.

The findings show that: jobchar1 (a 6 has been entered in at least one cell), burnout3 (again, a 6 has been entered in at least one cell), and itl2 (a 5 has been entered in at least one cell) have illegal codes; there are missing observations for every item with the exception of pe1, pe2, pe3, burnout10, itl1, and itl2.

To address the unauthorized entries, the proper measures were implemented. A closer look at the missing data showed that each participant provided either all or the vast majority of the questions.

Consequently, no surveys were discarded. Any missing data will not be considered in the analysis that follows.

From this point on, we can carry out more thorough analysis to evaluate the accuracy of our data.

Evaluating the efficacy of the measures

It is now possible to test the measures' validity and reliability.

RELIABILITY

Testing for both consistency and stability establishes a measure's reliability. The degree of coherence among the objects used to measure a notion is known as consistency. A reliability coefficient called Cronbach's alpha shows how well a group of items are positively associated with one another. Cronbach's alpha is calculated using the average intercorrelations between the concept-measuring items. The internal consistency reliability is higher when Cronbach's alpha is nearer 1. In certain circumstances, the split-half reliability coefficient is another metric for consistent reliability.

Based on how the scale is divided, different coefficients will be calculated because this represents the correlations between two half of a set of items. Occasionally, when evaluating many scales, dimensions, or factors, split-half reliability is acquired to test for consistency. The objects are divided according to predetermined logic across each of the dimensions or components (Campbell, 1976). An adequate test of internal consistency reliability is Cronbach's alpha in practically all cases. This chapter will demonstrate how computer analysis is used to determine Cronbach's alpha.

Parallel form reliability and test-retest reliability are two methods for evaluating a measure's stability. The existence of parallel form dependability is demonstrated by a good correlation between two comparable versions of a measure. Calculating the correlation between identical tests given at two distinct times allows for the establishment of test-retest reliability.

OUTSTANDING BUSINESSES : Evaluating the Multi-Item Measures Dependability

Every measure that used multi-item scales to measure perceived equity, burnout, work enrichment, and desire to leave requires a test of the respondents' responses to the scale questions for consistency. A well-liked test for inter-item consistency is Cronbach's alpha.

Reliabilities are often classified as good when they exceed 0.80, acceptable when they are between 0.70 and 0.60, and bad when they are less than 0.60. For the intention to leave measure, therefore, the internal consistency dependability of the measures utilized in this research can be deemed satisfactory, and the other measures.

It is crucial to remember that before submitting the questionnaire's items for reliability testing, all of its negatively worded items should be inverted. The reliabilities found will be inaccurate if not every item measuring a variable is pointing in the same direction.

Alternatively, we could utilize this table to determine which items would need to be eliminated from our measure in order to improve the inter-item consistency if our Cronbach's alpha was too low (below 0.60). Keep in mind that, often, eliminating an item has a negative impact on our measure's validity even while it increases its reliability.

The results on the original questions can now be aggregated into a single score since it has been determined that the inter-item consistency is adequate for perceived equity, job enrichment, burnout, and intention to leave. For example, the scores on the five distinct "perceived equity" items can be used to produce a new "perceived equity" score, but only after items 1, 2, and 4 have been reversed.

The scores on the four distinct "job enrichment" elements can also be used to construct a new "job enrichment" score, and so forth. As previously mentioned, this entails summing the scores for each participant or instance and dividing the result by the total number of items.

Validity

Factor analysis can be used to establish the factorial validity of the data. The multivariate method of factor analysis will validate whether or not the predicted dimensions show up. Remember from Chapter 11 that in order to operationalize the concept, measurements must first be defined by defining the dimensions. If the dimensions are, as proposed, tapped by the measure's items, factor analysis clarifies this. Testing the measure's ability to distinguish between people who are known to be different helps establish criterion-related validity (see Chapter 12's treatments of concurrent and predictive validity).

When two distinct sources respond to the same measure with a high degree of correlation (for example, when supervisors and subordinates react similarly to a perceived reward system measure given to them), convergent validity can be proven.

When there is no correlation between two clearly separate notions (such as courage and honesty, leadership and motivation, or attitudes and conduct), discriminant validity can be established. The multitrait multimethod matrix can be used to establish both convergent and discriminant validity. Books on factor analysis and the multitrait multimethod matrix are available for students who are interested in learning more.

It goes without saying that when well-validated measures are employed, there is no need to reestablish their validity for every study. It is possible to test the products' reliability, though.

PART 2 OF EXCELSIOR ENTERPRISES: DESCRIPTIVE STATISTICS

We may now move on to our data analysis after calculating the new ratings for perceived equity, job enrichment, fatigue, and intention to leave. For the multi-item, interval-scaled independent and dependent variables, descriptive statistics may now be derived, including variance, means, standard deviations, and minimum and maximum values. In addition, it is possible to generate a correlation matrix to investigate the relationships between the variables in our model.

This will assist us in determining the size of the issue and other crucial questions. Stated differently, how likely are Excelsior Enterprises employees to quit? What is the typical inclination to walk away?

What kind of issue does the situation have? Take a look at the histograms for example. In all scenarios, the average ITL is the same.

On the other hand, the graphs demonstrate that ITL is "fairly normally" distributed in the first hypothetical histogram.

We can also get answers to the following queries with the aid of descriptive statistics: Are the workers happy in their current positions?

What do the workers think about job enrichment?

To what extent are employees burnt out?

How many of them?

Does the degree to which employees believe their relationship with the organization is equitable vary much?

What connections exist between intention to leave, job enrichment, fatigue, perceived equity, and job satisfaction?

Along with the outcomes of our hypothesis testing, the answers to these questions will enable us to make well-informed decisions on the most effective way to address the issue.

Excelsior Enterprises' study yielded descriptive statistics for its interval-scaled independent and dependent variables, including variance, means, standard deviations, and minimums and maximums. It should be noted that every variable—aside from ITL—was rated on a five-point scale. There was a four-point rating system for ITL.

It is evident from the data that Excelsior Enterprises has a turnover issue, as indicated by the mean of 2.59 on a four-point scale for ITL.

When an employee's score is at least 1, it indicates that some do not plan to leave at all, and when it is at least 4, it indicates that some are seriously considering leaving. On a five-point scale, job satisfaction is a little below average (2.91). Both the mean for experienced burnout (2.57) and the mean for perceived equity (2.44 on a five-point scale) are relatively low. Lastly, there is a perception that the work is somewhat enriched (3.49). The variation for each variable is also very substantial suggesting that participant responses are not usually particularly close to the mean for each variable.

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