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Analysing and Presenting Quantitative Data: Inferential Statistics

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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).

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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.

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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.

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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

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Selection of statistical tests – examples Research questionIndependent variable Dependent variableStatistical 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? NominalRank order scoresSpearmans 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)

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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.

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Paired t-test data set

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Data outputs: test for normality Case Processing Summary Cases ValidMissing Total NPercentN N StressTime %11.1% % StressTime %11.1% % Tests of Normality Kolmogorov-Smirnov(a) Shapiro-Wilk StatisticdfSig.StatisticdfSig. StressTime StressTime a Lilliefors Significance Correction

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Data outputs: visual test for normality

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Statistical output Paired Samples Test Paired Differences dfSig. (2-tailed) MeanStd. Deviation Std. Error Mean 95% Confidence Interval of the Difference t LowerUpper Pair 1 Stress Time 1 Stress Time Paired Samples Statistics MeanN Std. Deviation Std. Error Mean Pair 1 StressTime StressTime

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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.

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Example of data gathering instrument

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Mann-Whitney U data set

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Statistical output Tests of Normality a Lilliefors Significance Correction Attitude Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed).000 a Grouping Variable: Sex Test Statistics(a) Ranks Sex Kolmogorov-Smirnov(a) Shapiro-Wilk StatisticdfSig.StatisticdfSig. Attitude Ranks SexNMean RankSum of Ranks Attitude Total 100 Ranks

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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.

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Chi-square data set

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Statistical output Chi-Square Tests Valuedf Asymp. Sig. (2-sided) Exact Sig. (2-sided) Exact Sig. (1-sided) Pearson Chi-Square.382(b)1.536 Continuity Correction(a) Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases 201 a Computed only for a 2x2 table b 0 cells (.0%) have expected count less than 5. The minimum expected count is Symmetric Measures a Not assuming the null hypothesis. b Using the asymptotic standard error assuming the null hypothesis. Value Approx. Sig. Nominal by Nominal Phi Cramer's V N of Valid Cases 201

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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.)

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Perfect positive and perfect negative correlations

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Highly positive correlation

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Strength of association based upon the value of a coefficient Correlation figureDescription None Negligible Weak Moderate Strong Very strong Perfect

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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).

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Associations between two ordinal variables For data that is ranked, or in circumstances where relationships are non-linear, Spearmans rank-order correlation (Spearmans rho), can be used.

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Spearmans rho data set

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Statistical output Correlations MrJonesMrsSmith Spearman's rhoMrJonesCorrelation Coefficient (**) Sig. (2-tailed)..000 N 30 MrsSmithCorrelation Coefficient.779(**) Sig. (2-tailed).000. N 30 ** Correlation is significant at the 0.01 level (2-tailed).

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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 Spearmans rho.

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Pearson Product-Moment data set

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Relationship between variables

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Statistical output Descriptive Statistics Mean Std. DeviationN Rainfall Sales Correlations RainfallSales RainfallPearson Correlation (**) Sig. (2-tailed).000 N 30 SalesPearson Correlation -.813(**) 1 Sig. (2-tailed).000 N 30 ** Correlation is significant at the 0.01 level (2-tailed).

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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.

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