Statistics for food
science ± VI: correlation
and regression (part B)
John A. Bower
Part A of ``Statistics for food science VI'' (SFS VIa) described the general characteristics and nature of correlation and regression. The current section looks at a selection of applications in food studies. Practitioners at all levels in research and industryapplythese techniques extensively. Examples range from the use of simple correlation and regression to more sophisticated instances using the multivariate regression methods of modern food research. Correlation analysis is the starting point in most cases and Pearson's coefficientris byfar the most common measure used.
Studies on food consumption and
nutrition
In the field of nutrition, researchers
continuouslygather vast amounts of data on food type and intake level. One of the intentions of the exercise, or ways of using this information, is to examine correlation patterns. Andersonet al.(1994) provide an example. Theyexamined dailyintake of fruit and vegetables (FVID) and its relationship with the intake of other foods (Table I).
All the coefficients are low and consequentlythe coefficients of
determination (SFS VIa) would also be low. Three of the coefficients are highlysignificant but the sample size, i.e. the number of subjects, is also verylarge. At first sight one would be forgiven for doubting the strength of the relationships shown, given the criteria explained in SFS VIa, but the data above, and especiallythe full data given in the paper (Andersonet al., 1994), indicate a marked trend. Namely, a higher fruit and vegetable intake seems to be associated with an increased consumption of chicken and a lower consumption of pies. Scatter diagrams may have provided more insight but these were not given or referred to in the work.
Coefficients based on smaller sample sizes also occur in nutrition studies. Drummondet al.(1998) studied eating frequencyrelated to bodyweight and to other measures (Table II).
Thervalues were still relativelylow but were of sufficient magnitude and significance (in the case of males) to contradict an existing hypothesis that postulated that a positive
The author
John A. Boweris a Lecturer in Food Science at the Faculty of Business and Consumer Studies, Queen Margaret University College, Edinburgh, UK.
Keywords
Statistical measures, Correlation analysis, Regression analysis
Abstract
Describes the application of correlation and regression in food science and nutrition studies.
Electronic access
The current issue and full text archive of this journal is available at
http://www.emerald-library.com
Nutrition & Food Science
Volume 30 . Number 2 . 2000 . pp. 80±85
relationship between frequencyand weight should exist.
Food studies can also involve measurement of consumer attitudes. Towler and Shepherd (1991/92) used correlation and regression to assess the association between attitudes and beliefs in relation to consumption of a high-fat food. The coefficients obtained ranged from 0.28 to 0.66. These values are higher but some confusion maystill exist regarding the ``required'' magnitude of correlation
coefficients in general.
How high should correlation coefficients be for importance?
Interpretation of coefficient size can be aided bya simple graded list (e.g. Morton et al., 1996; Table III). A more detailed account based on the ``effect size'' has been discussed
byCohen (1992). Values are quoted for Pearson's coefficient (Table III) when used with human behavioural data.
Care must be taken in implementation of these ``rules''. Practical consequences should be viewed as distinct from some statistical considerations when judging the importance of results. Provided that coefficients are significant (p< 0.05), then even a weak association of small effect size mayhave important practical consequences, or at least point to new directions for investigation. Sample size should be adequate for the purposes of the study(SFS VIa). Thus, for surveys of the type above, especially if based on random samples of the population, important information can be revealed bythe correlation analysis, even with low coefficients. This contrasts with the applications described below where coefficients are expected to be higher for acceptance.
Calibration, reliability and concurrence measures
Correlation can be used in a varietyof ways to gauge accuracyand reliability. Calibration occurs when measurements are performed on objects or ``standards'' of known magnitude. Thus, the measuring device (instrument or person) can be ``adjusted'' to these known levels on a scale, and accuracy(SFS II, Bower, 1995) will be improved. Reliabilityis an indicator of the consistencyof measurement, similar in concept to precision (SFS II). Concurrence is similar but is viewed as a guide to the level of agreement between measures.
Calibration in food analysis by instrumental measures
Manyanalytical methods relyon correlation and regression. Analysis of standards of known concentration produces the familiar ``calibration graph''. Unknown samples can then be ``read off'' the graph manuallyor calculated via the regression equation. The technique makes the usual assumptions of linearityand that there are no errors produced when the standards are made up and
analysed, i.e. there are no errors in theX variate. Small infringements of these
assumptions are likelyand these can become significant in current analytical methodology where detection limits are verylow. The error in both slope and intercept and the measured analyte can be calculated, enabling
Table IIPearson's correlation coefficients of daily eating frequency (EF) and body weighta
n r
EF vs:
Body weight (males) 42 ±0.34*
Body weight (females) 37 +0.14
Notes:
aAdapted from Table II of Drummondet al. (1998)
* Significant atp < 0.05
Table IPearson's correlation coefficients (r) of FVID and meal or snack intakesa
n r
FVID vs crisps 1407 ±0.082***
FVID vs chicken 1401 0.279***
FVID vs pies 644 ±0.214***
FVID vs red meats 1204 0.019
Notes:
aAdapted from Table I of Andersonet al. (1994)
*** Significant atp < 0.001
Table IIIInterpretation of the magnitude of a correlation coefficient
Magnitude ofr
Degree of
associationa Effect sizeb
0.8-1.0 Strong ±
0.5-0.8 Moderate Large
0.2-0.5 Weak Medium
0.1-0.2 Negligible Small
Notes:
confidence limits to be set for the determination (Miller and Miller, 1993). These procedures are more complex than the relativelysimple error measurement
procedures described in SFS II. Typically six calibration points are adequate, but veryhigh coefficients would be expected (r> 0.9) combined with a clear linear relationship.
Calibration in sensory evaluation
The same procedures can be used when the measuring instrument comprises a trained sensorypanel.
Food formulations of known composition or properties can be prepared as standards varying in some aspect related to sensory perception, e.g. odour compound
concentration, shear value, colour additive content, etc. The mean values for each sensoryattribute and instrumental measure are calculated. Correlation associations are done pair-wise, so if there are several variables then manycoefficients will be obtained.
Panel or panellist vs instrumental
Panel and panellist measures with a corresponding instrumental measure will reveal whether the panel or individual has achieved an acceptable level of accuracyand ``how good'' theyare at using a certain type of scale. This can be done at the level of ranking (Spearman's rho) and at interval scale level with Pearson'sr. The obvious extension is the use of reference samples or standards in sensoryresearch. The standards usuallyspan a range of values and can be used to calibrate the panel (Piggot, 1995). A simple test of this type is often used in selection programmes for sensorypanel members (BSI, 1993; ``Tests for discrimination between levels of intensity of stimulus''). Here, people with a reasonable abilityshould achieve perfect rank correlation or higher, equivalent to no more than one pair of adjacent samples out of sequence, e.g. 1 2 3 4 is in perfect sequence but 1 2 4 3 has two samples out of sequence. With a four sample set as specified in the BSI standard, this would mean a rank correlation of rs= 0.8 at
least, and 0.9 or higher with five samples. Some problems arise with calibration used in this way. One relates to certain assumptions which are discussed below. The other is that unlike instruments human panellists find it difficult to ignore other stimuli, although the calibration samples are usuallydesignated for a single specified stimulus. Linearitycan be improved bytransforming the instrumental
data before analysis. This is because of the psychophysical relationship being examined. Thus, according toFechner's Lawthe sensory response will be directlyrelated to the logarithm of the stimulus (Gacula and Singh, 1984), especiallyif the stimulus is stepped up in a ratio manner. Consequently, the linearity of sucrose concentration vs sensorysweetness intensitywill be improved if the logarithm of the concentration is taken before analysis. Beware that, while the latter result is feasible using sugar solutions, obtaining a similar relationship within a food product may demand more precise detection bypanellists.
Concurrence of sensory measures ± sensory vs sensory
These applications do not use instrumental data. Correlation of a panellist's mean value for a sensoryvariable, compared with that of the whole panel or another panellist, will indicate the degree of concurrence. Panellists with differing perception will have low positive correlations or even negative correlations with the panel mean or other members. This is undesirable in a trained panel and provides a guide for further training and selection procedures.
Comparison of panel means for one sensory variable with another is often done with sensoryprofiling data. This can identify measures that are possiblyredundant and can aid vocabularydevelopment, etc. during training and monitoring. Powers (1988) discusses these methods in general for sensory evaluation and Noronhaet al. (1995) give a specific example.
Accepting the drawbacks, correlation analysis can give useful guidance in sensory panel member selection, training and
monitoring of performance. Sample numbers range from 4-15 and coefficients are expected to be high (typicallyr> 0.7) but this depends on the application. Adequatelytrained panels maybe assumed to produce data with less ``noise'' (error) as detected bycorrelation and other methods of analysis, but checks should not be omitted. Theyprovide valuable screening and confirmation aids before further analysis.
General reliability of a measure
gauge attitudes, beliefs, or opinions, etc. A replicated or repeated measure should show high positive correlation if the measurement method is reliable. Alternativelytwo different measures which assess the same concept can be compared. Correlation as a reliability check should not be confused with a precision estimate such as the standard deviation (SFS II) which is applied to replicated measures on one object or sub-samples from one object. It is possible for data with poor precision within individual objects to show high correlation over a series of objects. Thus, in Table IV, correlation is high but the measures are well apart with poor precision as seen bythe extremelyhigh percentage coefficients of variation (SFS II).
One application is in ``test-retest'' reliability studies, where correlation can be used to check the stabilityof a measure over a period of time. Seamanet al.(1993) used correlation coefficients to establish whether or not consumers were reliable when measuring meat quality. A duplicate sample was
included in a set of six products tested by50 consumers. Correlation analysis of duplicates gave coefficients which were veryhigh (r> 0.76) and significant (p< 0.001). Scatter plots confirmed the linear association and within duplicate precision was also high as shown bylow scoring ranges for many consumers.
Relating sensory and instrumental data
In the field of food science relating sensory and physico-chemical data is a major application currentlyreceiving great attention. Its use in calibration has been discussed above but it has much wider application. The term ``sensory'' is usually taken to mean the descriptive profiling type, involving no hedonic or preference
indications, and ``instrumental'' refers to any chemical or physical determination. The method makes some assumptions: first that the measures are sufficientlyspecific and relevant to the task at hand, and second that the instrumental measure is more objective and is therefore ``correct''. Thus it gives the ``true'' values and contrasts with the more ``subjective'' nature of sensorymethods. These views have been questioned byseveral workers (Moskowitz, 1981; Piggot, 1998), one of the main problems being the
inadequacyor even irrelevance of one or both of the measures. For example, the use of an instrumental measure of texture that did not ``mimic'' the sensorymeasure adequately could result in a spurious correlation. Thus the distinction is debatable but it still underlies the basic rationale of both this application and that of calibration in sensory work.
Once researchers ascertain or assume validityand reliabilityof anysensoryand instrumental data theycan move with confidence to studythe information. Possible aims include elucidation of food properties and reactions or psychophysical relationships, or looking at the predictive power of the instrumental measure. Correlation is often the starting point with multivariate analysis methods such as principal component analysis, which utilises a correlation matrix of all variables, being prominent. Sample sizes can be larger than those used during panel training especiallyif the researchers are examining a range of market products. Gerbi et al.(1997) give a full scatter plot matrix of 15 sensorydescriptors applied to 96 wine vinegars. Joneset al.(1985) studied the correlation of sensoryand instrumental properties in 26 brands of beefburger. One example from the manycoefficients given shows the high correlation of sensoryand instrumental texture (Figure 1). The linear relationship (based on the logarithm of the instrumental measure) is obvious with the variates having a wide range on the scale used.
The authors used regression analysis to determine the instrumental variables which were best at predicting sensoryproperties (Joneset al., 1985). One of the features to look for is whether or not the scatter diagram reveals that the relationship is enhanced bya few points, rather than an overall trend across all the data. This is illustrated byPeppardet al.(1989) who were unhappywith the Table IVExample of data showing poor precision (within objects) and
high Pearson's correlation (all objects)
Object/subject Measure 1 Measure 2 %CVa
1 2 8 85
2 4 13 75
3 1 6 101
4 1 7 106
5 3 11 81
Correlation: M1 vs M2r = 0.987
separated points and wide scattering for one result in their studyon the relationship between chemical and sensorymeasures in beers (Figure 2).
A commerciallyvalued application deals with ascertaining the viabilityof using a rapid instrumental measure to predict some aspect of sensoryquality. This is often the ultimate objective of such studies and regression analysis is used to determine the predictive aspect. The advantage gained is based on the assumption that the instrumental measure can replace the more lengthysensory procedure, and avoid the error prone, ``noisy'' data generated bya sensorypanel. This is of particular interest to the food industryas it allows rapid, and hopefullyimproved, quality assurance methods. An instrumental method which shows high correlation with one or more sensorymeasures is a good candidate as a predictor of such sensoryproperties. Many instrumental measures can be performed in a
few seconds, for example, colour measures by tristimulus colorimeters, mechanical texture and rapid chemical analyses. Thus, a
regression equation for the relationship shown in Figure 2 would allow the instrumental test to be used as a predictor of ``rubberiness'' in burgers, etc. The use of instruments in this waycould implythat trained sensorypanels maybecome redundant and that the
instrument could be used as the sole measure.
Can instruments replace human panellists?
Despite the speed and convenience of instrumental analysis many food companies relyheavilyon sensorypanel input. One reason is that continual monitoring of any sensoryand instrumental relationship is required, especiallywhen product
formulations change or new products appear. Additionally, correlation and regression as applied above can be extended to include hedonic and preference measures, generated byconsumers. This is even more valuable to food manufacturers as theycan relate trained panel sensorydata and instrumental
measurements directlyto consumer
perception, and hence to sales potential. Two problems arise in this case. One is that the assumption of linearitymaybe infringed, therebyinvalidating the analysis (MacFie and Hedderley, 1993). Trantet al.(1981) cited examples of this malpractice and have illustrated it with several experiments. An adaptation of one of the scatter diagrams is shown (Figure 3).
The average hedonic response of two assessments, separated bya period of time, shows the nonlinear parabolic (inverted U) Figure 1Scatter diagram and correlation coefficient of
the relationship between sensory and instrumental texture in beefburger
Figure 2Scatter diagram showing isolated points. Prediction of sensory or affective quality via sensory or instrumental measures
shape of the curve. The linear correlation coefficients are low suggesting no
relationship, but a marked curvilinear function is present. Obviouslya straight line would not ``fit'' these data well and a curve fitting method would give a more valid equation of the relationship. The other difficultyrelates to the fact that individual responses maydiffer from the average and researchers must ensure that consumer data of this type are representative of the target population.
Regression applications
Manyresearch workers and food practitioners will use correlation analysis as an initial stage before regression. Some indications of this have been given in the examples above. In these studies and experiments simple
regression is rare and several instrumental and sensorymeasures maybe examined for their interrelationships. Thus multiple regression has wider application. There are several forms of multiple regression analysis available and partial least squares regression has been identified as the most promising and successful in its abilityto link consumer, sensoryand instrumental data (Piggot, 1998).
An illustration of correlation and regression will be presented in the final article (Part C).
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