Analysing and Presenting Quantitative Data: Inferential Statistics.

Analysing and Presenting Quantitative Data:

Inferential Statistics

Objectives

After this session you will be able to:

• Choose and apply the most appropriate statistical techniques for exploring relationships and trends in data (correlation and inferential statistics).

Stages in hypothesis testing

• Hypothesis formulation.• Specification of significance level (to see

how safe it is to accept or reject the hypothesis).

• Identification of the probability distribution and definition of the region of rejection.

• Selection of appropriate statistical tests.• Calculation of the test statistic and

acceptance or rejection of the hypothesis.

Hypothesis formulation

Hypotheses come in essentially three forms.Those that:

• Examine the characteristics of a single population (and may involve calculating the mean, median and standard deviation and the shape of the distribution).

• Explore contrasts and comparisons between groups.

• Examine associations and relationships between groups.

Specification of significance level – potential errors

• Significance level is not about importance – it is how likely a result is to be probably true (not by chance alone).

• Typical significance levels:– p = 0.05 (findings have a 5% chance of being untrue)– p = 0.01 (findings have a 1% chance of being untrue)

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Identification of the probability distribution

Selection of statistical tests –examples

Research question Independent variable

Dependent variable Statistical test

Is stress counselling effective in reducing stress levels?

Nominal groups (experimental and control)

Attitude scores (stress levels)

Paired t-test

Do women prefer skin care products more than men?

Nominal (gender) Attitude scores (product preference levels)

Mann Whitney U(data not normally distributed)

Does gender influence choice of coach?

Nominal (gender) Nominal (choice of coach)

Chi-square

Do two interviewers judge candidates the same?

Nominal Rank order scores Spearman’s rho(data not normally distributed)

Is there an association between rainfall and sales of face creams?

Rainfall (ratio data) Ratio data (sales) Pearson Product Moment (data normally distributed)

Nominal groups and quantifiable data (normally distributed)

To compare the performance/attitudes of two groups, or to compare the performance/attitudes of one group over a period of time using quantifiable variables such as scores.

Use paired t-test which compares the means of the two groups to see if any differences between them are significant.

Assumption: data are normally distributed.

Paired t-test data set

Data outputs: test for normalityCase Processing Summary

Cases

Valid Missing

Total

N Percent N Percent N Percent

StressTime1 92 98.9% 1 1.1% 93 100.0%

StressTime2 92 98.9% 1 1.1% 93 100.0%

Tests of Normality

Kolmogorov-Smirnov(a)

Shapiro-Wilk

Statistic df Sig. Statistic df Sig.

StressTime1 .095 92 .041 .983 92 .289

StressTime2 .096 92 .034 .985 92 .363

a Lilliefors Significance Correction

Data outputs: visual test for normality

Statistical output

Paired Samples Test

Paired Differences

df Sig. (2-tailed)

Mean Std. Deviation

Std. Error Mean

95% Confidence Interval of the Difference

tLower Upper

Pair 1

Stress Time 1Stress Time 2 1.60870 2.12239 .22127 1.16916 2.04823 7.270 91 .000

Paired Samples Statistics

Mean NStd.

DeviationStd. Error

Mean

Pair 1

StressTime110.3587 92 3.48807

.36366

StressTime2 8.7500 92 3.19555 .33316

Nominal groups and quantifiable data (normally distributed)

To compare the performance/attitudes of two groups, or to compare the performance/attitudes of one group over a period of time using quantifiable variables such as scores.

Use Mann-Whitney U.

Assumption: data are not normally distributed.

Example of data gathering instrument

Mann-Whitney U data set

Statistical outputTests of Normality

a Lilliefors Significance Correction

Attitude

Mann-Whitney U 492.500

Wilcoxon W 1020.500

Z -4.419

Asymp. Sig. (2-tailed) .000

a Grouping Variable: Sex

Test Statistics(a)

Ranks

Sex

Kolmogorov-Smirnov(a)

Shapiro-Wilk

Statistic df Sig. Statistic df Sig.

Attitude 1.298 32 .000 .815 32

.000

2 .167 68 .000 .909 68 .000

Ranks

Sex N Mean Rank Sum of Ranks

Attitude 132 31.89

1020.50

268 59.26

4029.50

Total 100

Ranks

Association between two nominal variables

We may want to investigate relationships between two nominal variables – for example:

• Educational attainment and choice of career.• Type of recruit (graduate/non-graduate) and

level of responsibility in an organization.• Use chi-square when you have two or more

variables each of which contains at least two or more categories.

Chi-square data set

Statistical outputChi-Square Tests

Value dfAsymp. Sig.

(2-sided)Exact Sig. (2-sided)

Exact Sig. (1-sided)

Pearson Chi-Square .382(b) 1 .536

Continuity Correction(a)

.221 1 .638

Likelihood Ratio .383 1 .536

Fisher's Exact Test .556 .320

Linear-by-Linear Association

.380 1 .537

N of Valid Cases 201

a Computed only for a 2x2 tableb 0 cells (.0%) have expected count less than 5. The minimum expected count is 33.08.

Symmetric Measures

a Not assuming the null hypothesis.b Using the asymptotic standard error assuming the null hypothesis.

ValueApprox.

Sig.

Nominal by Nominal

Phi.044

.536

Cramer's V .044 .536

N of Valid Cases 201

Correlation analysis

Correlation analysis is concerned with associations between variables, for example:

• Does the introduction of performance management techniques to specific groups of workers improve morale compared to other groups? (Relationship: performance management/morale.)

• Is there a relationship between size of company (measured by size of workforce) and efficiency (measured by output per worker)? (Relationship: company size/efficiency.)

• Do measures to improve health and safety inevitably reduce output? (Relationship: health and safety procedures/output.)

Perfect positive and perfect negative correlations

Highly positive correlation

Strength of association based upon the value of a coefficient

Correlation figure Description

0.00 0.01-0.090.10-0.290.30-0.590.60-0.740.75-0.991.00

NoneNegligibleWeakModerateStrongVery strongPerfect

Calculating a correlation for a set of data

We may wish to explore a relationship when:• The subjects are independent and not chosen

from the same group.• The values for X and Y are measured

independently. • X and Y values are sampled from populations

that are normally distributed.• Neither of the values for X or Y is controlled (in

which case, linear regression, not correlation, should be calculated).

Associations between two ordinal variables

For data that is ranked, or in circumstances where relationships are non-linear, Spearman’s rank-order correlation (Spearman’s rho), can be used.

Spearman’s rho data set

Statistical output

Correlations

MrJones MrsSmith

Spearman's rho MrJones Correlation Coefficient1.000

.779(**)

Sig. (2-tailed).

.000

N30

30

MrsSmith Correlation Coefficient.779(**)

1.000

Sig. (2-tailed).000

.

N 30 30

** Correlation is significant at the 0.01 level (2-tailed).

Association between numerical variables

We may wish to explore a relationship when there are potential associations between, for example:

• Income and age.• Spending patterns and happiness.• Motivation and job performance.

Use Pearson Product-Moment (if the relationships between variables are linear).

If the relationship is or -shaped, use Spearman’s rho.

Pearson Product-Moment data set

Relationship between variables

Rainfall70.0060.0050.0040.0030.0020.00

Sal

es

180.00

160.00

140.00

120.00

100.00

80.00

Statistical output

Descriptive Statistics

MeanStd.

Deviation N

Rainfall 48.17 11.228 30

Sales 132.47 28.311 30

Correlations

Rainfall Sales

Rainfall Pearson Correlation1

-.813(**)

Sig. (2-tailed)

.000

N 30 30

Sales Pearson Correlation-.813(**)

1

Sig. (2-tailed).000

N 30 30

** Correlation is significant at the 0.01 level (2-tailed).

Summary

• Inferential statistics are used to draw conclusions from the data and involve the specification of a hypothesis and the selection of appropriate statistical tests.

• Some of the inherent danger in hypothesis testing is in making Type I errors (rejecting a hypothesis when it is, in fact, true) and Type II errors (accepting a hypothesis when it is false).

• For categorical data, non-parametric statistical tests can be used, but for quantifiable data, more powerful parametric tests need to be applied. Parametric tests usually require that the data are normally distributed.