Chapter 3. Classification of Statistical Problems by Data Type. Chapter 3. Classification of Statistical Problems by Data Type

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1 Chapter 3. Classification of Statistical Problems by Data Type

2 Measurement Scales A measurement scale is the set of possible values that a variable can take on. We usually distinguish four levels of measurement scales. Remark: For the purpose of illustration we will usually take people to be the objects that are being measured. Each individual object (say person or car) is often termed a case in statistical packages. Nominal Scales On the nominal scale values of the variable are denoted by arbitrary numbers. For example suppose the variable is sex, then in measuring a female s sex we might record the number 1 and the number +1 for a male. In other words this lowest level of measurement consists of just category without order to these categories. Sex has no real numeric properties.

3 Ordinal Scales Ordinal scales are similar to nominal scales in that they specify categories, however there are the added feature that the categories can be ordered according to some property. Again this is non-numerical variable. For example, hospitals describe the condition of patients as 1 (resting comfortable), 2 (stable), 3 (guarded), and 4 (critical). These categories 1, 2, 3 and 4 are ordered but are not numeric. Note that the interval between categories 4 and 3 may not be equal to the intervals between categories 1 and 2. A common type of ordinal data is ranked data. That is we rank the n cases from 1 to n according to some criteria. Examples of ordinal variable are pain (none/moderate/severe), socio-economic status (low/middle/high), class of university degree, and other categorical variables that can be ordered.

4 Interval Scales Interval scales categorize, order, and quantify comparisons between pairs of measurements. Variables measured on interval scales are numeric variables. Interval scales have a defined unit of measurement and the difference between two interval level numbers is a measure of the difference in the property being measured. However, an interval scale does not have a natural zero, the zero position on the scale is arbitrary. The Fahrenheit and Centigrade scales for temperature are examples of interval scales. Because zero degrees, say Centigrade, does not mean no temperature we cannot say that 10 degrees centigrade is twice as hot as 5 degrees centigrade. Note that if we changed to the Fahrenheit scale the ratio would not be the same. However it is clear that the difference between 40 o F and 20 o F is the same as between 60 o F and 40 o F. Note, that because of a defined unit of measurement interval scales have equal intervals between successive values.

5 Ratio Scales The ratio scale is similar to the interval scale, but the position of zero is unique and indicates absence of the property. A ratio scale categorizes, orders, quantifies comparisons between pairs of measurements, and quantifies comparisons between individuals measurements. The Kelvin scale takes absolute as its unique zero (absence of heat) therefore the Kelvin temperature scale is a ratio scale. It is true that 40 o K is twice as hot as 20 o K. Length measure in feet, or miles, etc. is a ratio scale since it has a unit of measurement with an absolute origin, zero feet tall means absence of the property of height. In the ratio scale it is sensible to compare measurements by ratios, for example it is sensible to say that one person is twice as tall as another person. If it makes sense to say that one value is, say, k times another then the variable being measured is a ratio variable.

6 Remarks: The statistical analysis of Nominal variables is limited. Techniques such as frequency distributions, pie chart and bar chart may be constructed. Contingency table analysis may be useful with some chi-square tests, special measures of correlation between pairs of nominal variables may be calculated. A useful measure of center (a typical value) is the mode, but the median or mean makes no real sense. All the statistical techniques suitable for nominal data can be used on Ordinal data plus ranking and non-parametric procedures. A more extensive contingency table analysis is possible far sets of ordinal variables. Since the median takes order into account it is a useful measure of central tendency. But the mean is not a sensible number to calculate since the numbers assigned to the categories of an ordinal variable are arbitrary.

7 For interval variables all techniques useful on nominal and ordinal variables may be used, plus many more. For example one can include the calculation of the mean, the variance, the product-moment, the correlation and various parametric statistical calculations. Probably the most useful measure of central tendency is the mean. Finally all statistical techniques and calculations can be applied to variables measured on a ratio scale.

8 Qualitative vs Quantitative Variables Variables measured on either the nominal or ordinal scales are called qualitative variables. Variables measured on either the interval or ration scales are called quantitative variables. Discrete vs Continuous Variables Variables are discrete if they can assume either a finite or countable number of values. Discrete variables can be either qualitative or quantitative. Variables are continuous if they can assume any value in some interval. Continuous variables are quantitative. Remark: The discrete response number of accidents at a factory is quantitative (4 accidents is twice as many as 2 accidents). The discrete response pain measured on the ordinal scale (none, moderate or severe) is qualitative.

9 An Example[Furnace Data Set] The furnace data set describes a study of the effectiveness of two dampers used in gas furnaces to improve the heat loss through the chimney. Both dampers close the chimney when the furnace is in it off cycle. Damper type 1 (called EVD) closes electronically while damper type 2 (TVD) closes by means of a thermal switch which senses air temperature in the vent. A sample of 90 houses had 40 fitted with TVDs and 50 fitted with the EVDs. For each house the energy consumption was measured over several weeks and averaged with the damper working and without the damper working (i.e. with the damper in the chimney and without the damper in the chimney). Other factors that can affect energy consumption such as house size and type of gas furnace were measured and recorded.

10 Description of Furnace Data Variable type ch.area ch.shape ch.ht ch.liner house age btu.in btu.out damper Description of variable Type of furnace: 1=forced air, 2=gravity, 3=forced water Chimney area Chimney shape: 1=round, 2=square, 3=regtangujar Chimney height (in feet) Type of chimney liner: 0=unlined, 1=tile, 2=metal Type of house: 1=ranch, 2=two-story, 3=tri-level, 4=bi-level, 5=one and half stories House age in years (99 means 99 or more years) Average energy consumption with damper in the vent Average energy consumption with damper not in the vent Type of damper 1=EVD, 2=TVD

11 A copy of the actual data can be downloaded from the course website. We will use this data set to illustrate some of the statistical techniques useful for the analysis of nominal level variables. We will not adress, at tbis time, the central questioned posed by the power company concerning whether or not either damper was effective and which one was most effective. Note that the energy measurements are values of ratio variables. Nominal variables in the data set. Note that variables futype, chshape, chliner, housety, and damper are nominal variables.

12 Question 1: Which type of furnace, chimney shape, chimney liner and house is most popular? Question 2: Assume that the ranch houses are a simple random sample of ranch houses and that independently the two-story houses are a simple random sample of two-story houses. Are forced air furnaces less popular with home owners who live in two-story homes than with owners who live in ranch homes?

13 Solution 1: use R calculate the mode for each variable. Solution 2: To answer this question we must first determine the proportion p t, of two-story homes using forced air furnaces and p r, of ranch homes using forced air furnaces. We may calculate the two required proportions. We have the observed proportions ˆp t = 0.80 and ˆp r = It would appear that ranch home owners have a stronger preference for forced air gas furnaces than do two-story home owners. A natural way to compare the two population proportions is to calculate a confidence interval for the difference between them.

14 If n t, and n r are sufficiently large say n t min(ˆp t, 1 ˆp t ) > 5 and similarly for n r, then we may use the Central Limit Theorem to construct an approximate large-sample confidence interval for (p t p r ). We have (ˆp t ˆp r ) ± z α/2ˆσ where ˆσ = ˆp t (1 ˆp t )/n t + ˆp r (1 ˆp r )/n r. For a 95% confidence interval the above formula gives, 0.12 ± 1.96(0.7705) or ( , ). Since the confidence contains zero the data supports the hypothesis that the true population proportions are, in fact, equal at 95% significance level.

15 R Code ROW type ch.area ch.shape ch.ht ch.liner house age btu.in btu.out dam > # Reorganize the original data set "furnace.dat" and > # slice it into two files, "f1.dat" and "f2.dat" > # Replacing any "*" by NA > damper <- scan("f2.dat") Read 90 items > furnace <- read.table("f1.dat", header=t) > furnace <- data.frame(furnace, damper=damper) > furnace > > rm(damper) # Think about why we need to remove "damper"

16 R Code > attach(furnace) > > nt <- length(house[house==2]) > nt [1] 40 > ntf <- length(house[house==2 & type==1]) > ntf [1] 32 > pt <- ntf/nt > pt [1] 0.8 > nr <- length(house[house==1]) > nr [1] 38 > nrf <- length(house[house==1 & type==1]) > nrf [1] 35 > pr <- nrf/nr > pr [1] >

17 R Code > sigma <- sqrt(pt*(1-pt)/nt+pr*(1-pr)/nr) > sigma [1] > qnorm(0.975, 0,1) [1] > za <- qnorm(0.975, 0,1) > za [1] > (pt-pr)-za*sigma; (pt-pr)+za*sigma [1] [1] > # Check the conditions > nt*min(pt, (1-pt)); nr*min(pr, (1-pr)) [1] 8 [1] 3