Quantitative Methods in Managment. An introduction to GLMs and measurement theory

Transcription

1 Quantitative Methods in Managment An Introduction to GLMs and measurement theory Graeme D. Hutcheson 1 Luiz Moutinho 2 1 School of Education Manchester university 2 Department of Management Glasgow University This course is structured around Generalized Linear Models - a powerful system for analysing data. This course provides a basic general introduction to statistical analysis and concentrates on the types of data and hypotheses common in the social sciences. As no attempt has been made to hide the underlying theory or complexities associated with the techniques, on first inspection, it may look quite complex. This course is, however, MUCH EASIER THAN IT LOOKS.

2 Traditional statistics courses in social science are based around teaching a number of seemingly separate tests. For example, t-tests, ANOVA, MANOVA, MANCOVA, Mann-Whitney, Kruskal-Wallis, Friedman, chi-square, correlation and regression. An example data set Subject Gender Age Nationality EconStatus FactorSocial AgreeQ08 ManGrade subject01 male 23 Russian 2ses NA 1stAgree 1junior subject02 male 24 English 4ses neither NA subject03 female 31 Welsh 1ses agree 1junior subject04 male NA NA NA NA 5stDisagree 3upper subject05 female 43 Irish 2ses agree 2middle subject06 female 41 German 2ses NA 4disagree 2middle subject07 female 19 German 3ses agree 3upper subject08 female 38 Portuguese 3ses neither 2middle subject09 male 59 Spanish NA NA NA subject10 female 24 Scottish 2ses NA NA subject11 male 39 Irish 4ses stDisagree 1junior subject12 male 22 Irish 3ses agree NA subject13 female 64 Japanese 5ses disagree 3upper See ExampleData.csvHutcheson at and Moutinho

3 Traditional analyses applied to the example data set Compare the ages of males and females: t-test Compare the ages of the different nationalities: ANOVA Compare the Q08 agreement scores for males and females: Mann-Whitney Compare the Q08 agreement scores for the different management grades: Kruskal-Wallis The relationship between FactorSocial and age: correlation/regression The relationship between nationality and gender: Chi-square Note: A whole set Hutcheson of different and Moutinho testsan apply introduction for to repeated-measures GLMs and measurement theory or matched-sample designs. This form of statistical education is not ideal for a number of reasons... It does not give a theoretically-unified method for data analysis It does not allow appropriate tests to be easily identified We cannot easily add extra variables to the statistical tests. It does not provide a simple path for students to progress to other techniques

4 The Generalized Linear Model GLMs provide a superior method for analysing data. One that is modern, more poweful, easier to apply and easier to learn and remember. The idea behind them is simple... A particular variable (the response, Y) may be predicted using other variables (the explanatories, X). Y X 1 + X 2 + X 3 Share Price may be predicted by Output and Market Confidence. Marks may be predicted by Gender To run a GLM, we just need to identify the variable we wish to model and the information we are going to use to model it. If the variable being predicted is numeric, the GLM model is OLS regression (this technique includes the t-tests, ANOVAs, ANCOVAs, etc.,) If the variable being predicted is ordered, the GLM model is the proportional-odds logit model (this technique replaces Mann-Whitney, Kruskal-Wallis, Friedman, etc.,) If the variable being predicted is unordered categorical, the GLM model is the multinomial logit model (this technique includes the chi-square tests) Note: the same tests also apply for repeated measures designs.

5 The example data set (again) Subject Gender Age Nationality EconStatus FactorSocial AgreeQ08 ManGrade subject01 male 23 Russian 2ses NA 1stAgree 1junior subject02 male 24 English 4ses neither NA subject03 female 31 Welsh 1ses agree 1junior subject04 male NA NA NA NA 5stDisagree 3upper subject05 female 43 Irish 2ses agree 2middle subject06 female 41 German 2ses NA 4disagree 2middle subject07 female 19 German 3ses agree 3upper subject08 female 38 Portuguese 3ses neither 2middle subject09 male 59 Spanish NA NA NA subject10 female 24 Scottish 2ses NA NA subject11 male 39 Irish 4ses stDisagree 1junior subject12 male 22 Irish 3ses agree NA subject13 female 64 Japanese 5ses disagree 3upper See ExampleData.csvHutcheson at and Moutinho GLM analyses of the example data set Compare the ages of males and females: Model: Age Gender Test: OLS regression Compare the ages of the different nationalities: Model: Age Nationality Test: OLS regression Compare the Q08 agreement scores for males and females: Model: AgreeQ08 Gender Test Proportional-odds Compare the Q08 agreement scores for the different management grades: Model: AgreeQ08 ManGrade Test: Proportional-odds The relationship between FactorSocial and age: Model: FactorSocial Age Test: OLS regression The relationship between nationality and gender: Model: Nationality Gender Test: Multinomial regression

6 Numeric, ordered and unordered data can be analysed using just three regression techniques. They incorporate or replace the traditional tests... As many explanatory variables can be added as needed... They allow diagnostics and model improvement. Many other techniques are based on these (for example, survival, Rasch, multi-level, random-effect, structural equation models etc.,) It is easy to run the models...

7 Types of data Although a number of different schemes have been proposed that utilise a variety of categories and sub-divisions, we will only distinguish between three distinct scales of measurement unordered categorical, ordered categorical and numeric, based on a simplified version of Stevens 1946 classification of measurement scales. It is important to identify these three measurement scales as they are qualitatively different, are coded using distinct methods, and represent what is, perhaps, the minimum required for any course that aims to provide a general introduction to data analysis.

8 1: Unordered categorical scales An unordered categorical categorical scale of measurement is achieved when the data are recorded as categories which have no meaningful order. The only information provided is the category identifier. This scale is also known as a classificatory scale and a labelling system. Examples of unordered categorical data Car Treatment Manufacturer Gender Group Subject 1 Ford male A subject 1 2 Citroen male A subject 2 3 Volvo female A subject 3 4 Volvo male B subject 4 5 Renault female B subject 5 6 Land Rover female B subject 6 7 Toyota female C subject 7 8 Toyota male C subject 8 9 Volvo male C subject 9 10 Chrysler female C subject 10

9 Operations that can be applied to data recorded on an unordered categorical scale Unordered operations: most frequent car is Volvo (n=3) same numbers of male and females (n=5) Analysing data recorded on an unordered categorical scale As the only information we have about the variables is category membership, analytical techniques for unordered categorical data require techniques that only take into account category membership (for example, chi-square, multinomial logistic regression and log-linear analyses). Statistics designed for ordered or numeric data CANNOT be used to model these data. (If you are able to compute these statistics, your data have been wrongly coded!)

10 2: An ordered categorical scale of measurement is achieved when the data are recorded as categories that can be arranged in order according to some criteria. The only information provided is the category identifier from which an order can be established. Examples of ordered categorical data Highest Agreement Examination Mental Health Qualification Rating Grade Rating 1 A level strongly agree B no symptoms 2 O level disagree A impaired functioning 3 Masters neither A no symptoms 4 Degree disagree C mild symptoms 5 Degree agree D moderate symptoms Variable order: Highest Qualification: No qualification, O-level, A-level, degree, master, doctorate Agreement Rating: strongly agree, agree, neither, disagree, strongly disagree Examination Grade: A, B, C, D, E. Mental Health: no symptoms, mild symptoms, moderate symptoms, impaired functioning.

11 Operations that can be applied to data recorded on an ordered categorical scale Unordered operations: mental health: no symptoms is the most frequent Examinations: A grades are the most frequent Ordered operations: agreement: strongly agree > neither educational achievement: Degree < Masters Analysing data recorded on an ordered categorical scale Ordered categorical data may be analysed using techniques designed for unordered categorical data (eg., multinomial logistic regression). Ordered categorical data can also be analysed using techniques which take account of order in the categories (for example, proportional-odds logit models, Mann-Whitney, Friedman). Statistics designed for numeric data CANNOT be used to model these data (if you can compute these, your data are miscoded).

12 3: A numeric scale of measurement is achieved when the recorded data can be considered to have a direct relationship with the variable being measured. At a very basic level, the data provides information about the actual magnitude (size, quantity, distance, etc.,) of the information. In other words, the numbers representing the variable have substantive meaning. Examples of numeric data Temperature Daily gas Examination Hourly rate difference consumption result of pay m 3 62 % m 3 73 % m 3 58 % m 3 37 % m 3 71 % m 3 59 % m 3 63 % m 3 63 % 5.80

13 Operations that can be applied to data recorded on a numeric scale Unordered operations: most frequent hourly pay = 5.80 (n=4) most frequent examination results = 63% (n=2) Ordered operations: 7.98 > m 3 < 74 m 3 Numeric operations: 74m m 3 = 143m 3 73% 61% = 12% Analysing data recorded on a numeric scale Numeric data may be analysed using techniques designed for unordered and ordered categorical data (eg., multinomial logistic regression and proportional-odds logit models). Numeric data can also be analysed using techniques which take account of the magnitude of differences between observations (for example, OLS regression, t-tests, ANOVA and ANCOVA).

14 The scale of measurement used to record information about a particular variable is very important, as it is this that defines the statistical analyses that may be applied. Measurement theory encourages people to think about the meaning of their data. It encourages critical assessment of the assumptions behind the analysis. It encourages responsible real-world data analysis Ṡarle, 1995 It is important to not only identify the appropriate measurement scale, but to also use an appropriate coding scheme... Coding

15 Coding Coding aims There are many rules and conventions that can be applied to coding data. The following are a few that can be applied as general rules for data coding. The main principles are that coded data should... accurately represent the measurement scales and not contain more information than is actually available (i.e. DO NOT use numeric codes to indicate categorical data). represent information without the use of any hidden codes or labels. be coded clearly and unambiguously. be of a form that can be easily imported into different software packages (the coded data should be transportable). Coding An example data frame

16 GLMs provide a powerful method for analysing data. Different GLMs are applied to different measurement scales. It is important that these scales are identified and accurately represented. To enable data-sharing and transparency, the datasets used in this course are coded in ASCII and saved using the standard comma-separated-variable format (.csv).