BA Management Studies

Research Methods

Lecture 8:

Quantitative data analysis

Alexandra Dias

University of Leicester School of Management

Overview of the Lecture

Making sense of Quantitative Data

• Bivariate analysis

• Multivariate analysis

Quantitative data analysis

• Statistical significance: Testing procedure

• Types of error

• Tests of statistical significance

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Bivariate analysis(analysis of two variables at a time)

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Bivariate analysis

• Explores relationships between variables

• Searches for co-variance and correlations

• Cannot establish causality

• Can sometimes infer the direction of a causal relationship

• Contingency tables

– connects the frequencies of two variables

– helps you identify any patterns of association

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Pearsons r: the relationship betweentwo 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

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Pearsons r: example

Relation between credit rating class and probability of default

(PD). Is it linear? Can we use r?

If: 1=AAA,2=AA,...,9=C then r(class, Ln(PD)) = 0.995

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Pearsons r: the relationship betweentwo interval/ratio variables

• 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

– here we are assuming causality!

In the following example causality is justified by financial

theory and not by statistical analysis

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Coefficient of determination: example

Financial Theory says that stock returns can be explained by

the financial market returns

Assume that the market returns are given by the S&P 500

index and inspect if these explain the returns on the IBM

stock.

• Dependent variable: IBM

returns

• Independent or explanatory

variable: S&P 500

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Coefficient of determination: example

Data: Daily returns on closing prices from 1 Jan 2002 to 1 Jan 2012.

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Analysing the relationships betweenother, or mixed types of variables

• Spearmans 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)

• Cramers V: for the relationship between two nominal

variables, or one nominal and one ordinal variable (values

between 0 and 1)

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Analysing the relationships betweenother, or mixed types of variables

• 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 (nominal and interval/ratio), even when there is

no linear relationship between them

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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

Ex: correlation between stocks on UBS CH (bank) and

stocks on a Pensions Fund managing firm in UK could be

moderated by global financial market performance

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Quantitative Data Analysis

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Statistical significance

• How confident can we be that the findings from a sample

can be generalized to the population as a whole?

• How risky is it to make this inference?

• Only applies to probability samples

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Statistical significance of the mean:Gym visitors example

We can be 95% confident that the population mean (the

unknown real mean) is between

33.55+1.96×1.04 = 31.5 and 33.55−1.96×1.04 = 35.6

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Testing procedure for statisticalsignificance

1. Set up a null hypothesis: suggesting no relationship

between examined variables in the population from which

the sample was drawn;

2. Decide on an acceptable level of statistical significance;

Usually 5% or 1%

3. Use a statistical test;

4. If acceptable level attained, reject null hypothesis;

If not attained, do not reject it.

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IBM and S&P 500 example

Return(IBM) = 0.00018 + 0.82 × Return(S&P500) but 0.00018 is not significant(p > 0.05) hence Return(IBM) = 0.82 × Return(S&P500)

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We might be wrong to accept orreject the null hypothesis

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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

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The chi-square test

• The chi-square (χ2) 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.

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The chi-square test (cont.)

• 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.

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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)

Ex: if coefficient is 0.62 and p < 0.05, we can reject the

null hypothesis of no correlation

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Comparing means

• Statistical significance of relationship between two variables

means

• Total variation in dependent variable:

– error variance (variation within subgroups)

– explained variance (variation between subgroups)

• F statistic

– expresses amount of explained variance in relation to

amount of error variance

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References

• Brymann and Bell, 2011, Business Research Methods, 3rd

Ed, Oxford University Press

2 Chapter 14.

• Easterby-Smith, Thorpe and Jackson, 2008, Management

Research, 3rd Ed, Sage

2 Chapter 11.

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