Social Research Methods Alan Bryman Social Research Methods Chapter 15: Quantitative data analysis Slides authored by Tom Owens

Introduction Think about data analysis at an early stage in the research process Decisions about methods and sample size affect the kinds of analysis you can do Page 330 2

Types of variable Interval/ratio Ordinal Nominal Dichotomous regular distances between all categories in range Ordinal categories can be ranked, but unequal distances between them Nominal qualitatively different categories - cannot be ranked Dichotomous only two categories (e.g. gender) Page 335 3

Deciding how to categorize a variable Figure 15.1 Page 336 4

Univariate analysis (analysis of one variable at a time) Frequency tables Number of people or cases in each category Often expressed as percentages of sample Interval/ratio data need to be grouped Diagrams Bar chart or pie chart (nominal or ordinal variables) Histogram (interval/ratio variables) Page 337 5

A bar chart (gym study) Figure 15.2 Page 338

A pie chart Main reasons for visiting the gym Figure 14.3 Page 344

A histogram Page 15.4 Page 338

Measures of central tendency Mean Sum all values in distribution, then divide by total number of values Median Middle point within entire range of values Not distorted by outliers Mode Most frequently occurring value Page 338, 339 9

Measures of dispersion Dispersion means the amount of variation in a sample. Measures of dispersion compare levels of variation in different samples to see if there is more variability in a variable in one sample than in another. The range is the difference between the minimum and maximum values in a sample The standard deviation is the average amount of variation around the mean, reducing the impact of extreme values (outliers) Page 339 10

Bivariate analysis (analysis of two variables at a time) Explores relationships between variables Searches for co-variance and correlations Cannot establish causality Can sometimes infer the direction of a causal relationship If one variable is obviously independent of the other Contingency tables Connects the frequencies of two variables Helps you identify any patterns of association Page 340, 341 11

Pearson’s r : the relationship between two interval/ratio variables Coefficient shows the strength and direction of the relationship Lies between -1 (perfect negative relationship) and +1 (perfect positive relationship) Relationships must be linear for the method to work, so, plot a scatter diagram first Coefficient of determination Found by squaring the value of r Shows how much of the variation in one variable is due to the other variable? Page 342, 344 12

Analysing the relationships between other, or mixed types of, variables Spearman’s rho: for the relationship between two ordinal variables, or one ordinal and one interval/ratio variable (values of -1 to +1) Phi coefficient: for the relationship between two dichotomous variables (values of -1 to +1) Cramer’s V: for the relationship between two nominal variables, or one nominal and one ordinal variable (values between 0 and 1) Comparing means: when a nominal variable is identified as the independent variable, the means of the interval/ratio variable are compared for each sub-group of the nominal variable eta: for the level of association between different types of variables, even when there is no linear relationship between them Page 344, 345 13

Multivariate analysis (three or more variables) The relationship between two variables might be spurious Each variable could be related to a separate, third variable There might be an intervening variable A third variable might be moderating the relationship e.g. correlation between age and exercise could be moderated by gender Page 345, 346 14

Example of a spurious relationship Figure 15.11 Page 345 15

Statistical significance How confident can we be that the findings from a sample can be generalised to the population as a whole? How risky is it to make this inference? Only applies to probability samples Page 347 16

Testing procedure for statistical significance Set up a null hypothesis - suggesting no relationship between examined variables in the population from which the sample was drawn; Decide on an acceptable level of statistical significance; Use a statistical test; If acceptable level attained, reject null hypothesis; if not attained, accept it. Page 347, 348

…but we might be wrong to accept or reject the null hypothesis Type I and Type II errors Figure 15.12 Page 349 18

Tests of statistical significance The chi-square test establishes how confident we can be that there is a relationship between the two variables in the population Correlation and statistical significance provides information about the likelihood that the coefficient will be found in the population from which the sample was taken Comparing means and statistical significance – the F statistic expresses the amount of explained variance in relation to the amount of error variance Pages 348, 350

The chi-square test The chi-square (2) test is applied to contingency tables. It establishes how confident we can be that there is a relationship between the two variables in the population. The test calculates for each cell in the table an expected frequency or value - one that would occur on the basis of chance alone. The chi-square value is determined by calculating the differences between the actual and expected values for each cell and then summing those differences. Whether a chi-square value achieves statistical significance depends not just on its magnitude but also on the number of categories of the two variables being analysed. This latter issue is governed by what is known as the ‘degrees of freedom’ associated with the table. Page 355

Correlation and significance How confident can we be about a relationship between two variables? Whether a correlation coefficient is statistically significant depends on: the size of the coefficient (the higher the better) the size of the sample (the larger the better) e.g. if coefficient is 0.62 and p<0.05, we can reject the null hypothesis Page 355 21

Comparing means Statistical significance of relationship between two variables’ means Total variation in dependent variable: error variance (variation within subgroups of IV) explained variance (variation between subgroups of IV) F statistic expresses amount of explained variance in relation to amount of error variance Page 356 22