T h e analysis of the d a t a of an experiment has proceded along rigor
ous lines, b u t it should be emphasized t h a t t h e interpretations and significances pertain only t o t h e d a t a on hand. However, t h e experi
menter is usually particularly interested in extending t h e interpretations to future results. T h e extension is delicate. As an analogy, if one were given a sample of an ingot or of a carload of metal t o analyze, no m a t t e r how carefully t h e analysis of t h e sample were made, t h e results of the analysis would still be valid only for the sample; t h e extension of the d a t a t o the ingot or carload would depend entirely on the extra-experi
mental evaluation of the representation of the sample. Unless one knew something about t h e uniformity of the ingot or t h e homogeneity of the contents of t h e carload t h e extension is quite meaningless. T h e present situation is quite similar. T h e extension of significance t o future experi
mental results depends entirely on t h e judgment of the spectrographer.
A few guide lines are available. I t is known t h a t t h e s t a n d a r d deviation of a m e a n square, σ0 2, as obtained in t h e analysis of variance with / degrees of freedom m a y be estimated as follows;
α·(σο2) = yjj σ ο 2
This m a y be too high in t h e general case for a n y precise prediction unless / , t h e n u m b e r of degrees of freedom, is large. Hence, t h e use of small samples is still unsatisfactory for prediction, even though t h e analysis of results already obtained is on a sound basis. This follows, because if one were interested only in t h e results already on hand, one could consider t h e m as reliable a sample of a hypothetical population as one desires.
Another difficulty presents itself in t h e estimation of overall precision, i.e., t h e t o t a l m e a n square. A factorial experiment m a y have been performed over factors of varying orders of significance or importance.
For example (Table I I ) , t h e experiment P4QZR2S* yields a t o t a l m e a n , ± Total Sum of Squares _ A _ . . . . square equal t o ^ B u t suppose Ρ were a highly significant effect a n d R of very low significance. T h e n if an experiment of form P2Q*R*S6 were performed, still with 144 observations, t h e total sum of squares would be expected t o be m u c h less, consequently t h e
_ . . . . ... . . , Total Sum of Squares total m e a n square, which would still be equal t o ^ might well be significantly less t h a n t h e first value. This difficulty
m a y be avoided by t h e use of variance components, column 4 in Table I I . T o predict t h e results of a variance analysis, based on P4Q3P2£6, of an experiment of another form, P^ffiS1, one m a y compute t h e various variance components of t h e first experiment, i.e., σ2 0, a2PQS, a2PQ, a2PS, σ2^ , σ2Ρ, σ2ο, σ25, and then combine t h e m using coefficients derived from (i,j, k, I) instead of (4, 3, 2, 6) to yield a mean square: σ2 = σ2 0 + ka2PQs + kh2PQ + kja2PQ + ika2QS + ijka2s + ikh2Q + jkla2P. I t is a p p a r e n t how this pro
cedure m a y be used to redesign a new experiment t h a t would emphasize t h e factors previously determined to be significant.
T h e application of this procedure t o experiments involving regression is more involved. As has been stated, t h e variance of a single deter
mination, where regression is involved, m a y be written as
i.e., a2yi is the sum of the total variance reduced for regression, plus t h e variance of the means plus Xi2 times t h e variance of regression coefficient, or the variance of reg. This form shows the danger of extrapolation, since the factor x^ increases rapidly with xiy i.e., the distance between Xi a n d t h e mean. I n order to predict an average or over-all variance, one m a y use the form
where Μ is the m a x i m u m xiy ( since ^ / x2dx = -p- J. Since in t h e use of standards there is an inversion in the roles of Y and X one m a y compute
where b is the slope of Y on X. T o estimate the s t a n d a r d error of a new experiment one m a y pool the error due to deviations as a non-reducible error a n d compute t h e changes in the errors of the means and reg accord
ing to the new sets of levels using the d a t a as presented in Table V. I n general, t h e s t a n d a r d errors will accumulate so rapidly, if the effects are significant, t h a t the estimate will hardly be precise or close enough for practical use. A more practical procedure would be to arrange s t a n d a r d izing experiments in the form in which the actual analyses are performed, for example with one replicate and one excitation condition (including electrode form) on one plate with m a n y replications over m a n y plates in one day and over m a n y days.
However one condition peculiar to analytical work m a y be discussed, namely t h e use of repetitive measurements. I t is, or has been, a rather
Σχ2
1
commonly-spread notion t h a t analyses in duplicate, or triplicate will decrease t h e error of t h e m e a n of the repeated observations t o one-half or one t h i r d t h e error of a single observation. This notion m a y now be recognized as false. Considering t h e statistical analysis into variance components, it can be seen t h a t increasing the levels of P , i.e. using R*
instead of R1, will merely alter t h e value of σ0 2; t h a t is t h e new non
reducible error is σ0 / 2 = <r0 2/3. Indeed the effect on the error variance of repetition or replication to a n y extent can be predicted b y t h e use of appropriate divisors in the formula exhibiting the total error variance as the sum of its various components. I t has been widely observed t h a t R (Table I I ) for spectrographic procedure is almost always t h e least significant of all effects; hence decreasing it will affect almost no increase in precision. On the other hand, if the observations were truly replicated entirely, t h a t is, repeated observations m a d e with new plates, new excitation conditions (although nominally t h e same), new electrodes, new densitometer readings, etc., t h e n the mean of replicated readings would have an error reduction proportional t o t h e square root of the n u m b e r of observations, i.e., σ0' = χ/σ02/η. Mandel, has intro
duced a measure for this relationship a n d called it t h e coefncient-of-improvement, (CI)N. I t is defined as follows (CI)Ν = n' and means the following: n' is the n u m b e r of true replicates required t o give t h e improvement in precision obtained b y using Ν repetitive observations.
T h e ratio n'/N m a y be called the efficiency of improvement. T o one who has not measured these quantities, the low efficiency of improve
m e n t by repetitive observations, in spectrographic analyses at least, m a y be very surprising. Some numerical values will be given in the following section.
An i m p o r t a n t use of variance components is in t h e estimation of t h e expected improvement b y changing experimental conditions. T h u s if Q = electrode form, and σ0 2 is highly significant, it would be a p p a r e n t t h a t an investigation into t h e closer control or a more suitable electrode form would be likely t o bring about a m a r k e d improvement.
Estimations of precision are based on t h e formula
d — ta
or if the degrees of freedom are sufficiently high,
d = ca
where d is t h e interval extended in b o t h directions from t h e mean or true value which contains t h e proportion of the population of values indicated b y t h e probability level of t. Or, d m a y be defined as t h e least difference between single observations which would be considered
sig-nificant at the indicated probability level; σ is the s t a n d a r d error of t h e population. If, instead of single observations, t h e means of η replicated observations (not repeated ones) were considered, the least significant difference would be d = t
\/n
Tables m a y be prepared, based on the experimental value of σ, arranged either for a scale of values of d (say 1%, 5 % of content, etc.) or for a scale of probability values (say 9 5 % , 9 0 % , 8 0 % etc.).
Least significant differences m a y be also be used as an aid in t h e analysis of variance in t h e following manner. Suppose, in t h e analysis Table I I , it develops t h a t Ρ is a significant effect and it is desired t o locate just w h a t plate, or plates, are the main contributor t o t h e sig
nificance even though the variances are homogeneous. Since t h e experi
m e n t is of t h e form P4Q3#2£6, there are 3 X 2 X 6 = 36 observations on each plate, and t h e standard error of a plate mean is σ/ - \ / 3 6 = σ/6, and t h e s t a n d a r d error of a difference between plate means is σ' = Λ/2 ^·
Hence t h e plate means are arranged in increasing order, for example.
Pi, P2, P3, P4 and t h e least significant difference computed as d' = ta where a is the selected significance level. One can then
deter-6
mine, b y comparison of the observed differences with d', which plates are significantly different. T h e procedure may, of course, be applied within any effect or interaction.
I t should be noted, t h a t since in t h e regression analysis, the curve was computed as a regression of Δ log I on log per cent, for actual use it is necessary t o invert the role of independent and dependent variables.
Hence if b is the slope of the regression line as computed, 1/6 is the slope of the working curve (log per cent = F , Δ log I = X) and the above formula becomes d = t -4^·
b
1.11. Conclusion
Before considering some numerical examples, it should be pointed out how easily one m a y arrive at misleading estimates of precision pre
pared in an intuitive manner. T h e use of too small a sample as a basis for estimation is a common fault; for example, an analytical curve based on three standards, or three plates (notoriously a non-uniform factor) and so on. I t is also difficult t o see how quantitative measures of the contributions of various factors t o t h e total error can be easily estimated, for statistical significance is quite independent of physical significance or importance. A procedure with low total physical error m a y still have statistically significant effects, and vice versa,
It is the opinion of t h e writer t h a t in establishing an analytical curve, for example, great care should be t a k e n t o h a v e it based on m a n y observa
tions. Spot or daily checks should be considered only as indications whether t h e conditions or factors of t h e experiment are in normal control, allowing for fluctuations t o be expected of small samples. If t h e check results are not satisfactory, t h e experimental conditions should be investigated for sources of error or lack of control and t h e results of a few observations should not be used as t h e basis for a new analytical curve. However, even this s t a t e m e n t should not be considered too restric
tive. F o r example, if plates are found t o be a very highly significant source of variations, even a spot adjustment for each plate, poor as it m a y be, m a y still effect a significant improvement in precision. How
ever, such a condition suggests t h e need for closer investigation t o insure greater stability.
A final caution should be given before considering a computation.
T h e discontinuous n a t u r e of measurement has nothing t o do with the
" l a w of error," which is t h e specification of t h e parent population. T h e step or least count of t h e instrument, being finite, simply has t h e effect of grouping t h e observations into class intervals. Such a grouping m u s t always be accomplished before a frequency curve, or frequency polygon can be constructed; if t h e instrument did not a t t e n d to this, the computer would h a v e t o do it.
I t might be expected t h a t the computed values of η measurements would v a r y somewhat as t h e least count and zero of the measuring scale are changed, a n d such, in fact, is t h e case. This effect has been carefully investigated b y statisticians and corrections t o be applied t o t h e various derived quantities on account of the finite width of the class intervals h a v e come t o be known as "corrections for c o n t i n u i t y " or " S h e p p a r d ' s corrections." (These are not t o be confused with t h e transformation used t o analyze a sequence of measurements t h a t are merely ordinal classifications, a t y p e of observation usually not met with in spectro
graphic data.) T h e corrections serve t o bridge t h e gap between t h e continuous law of error a n d the discontinuous n a t u r e of measurement.
Such investigations have served t o show t h a t the least count of the instru
m e n t should be small enough so t h a t when a large n u m b e r of readings (well over 100) are taken, there will be a variation in the significant terminal digits of around 20 units, for otherwise a considerable portion of a set of observations is, in effect, scrapped. An astonishingly large n u m b e r of observations m a y be required t o overcome the damage done by unnecessarily coarse reading or graduation of the scale, or else t h e handicap of t h e coarse scale if a finer scale is not possible, in an experi
mental sense. Unfortunately, suitable continuity corrections have not
been devised for t h e analysis of variance; hence t h e computer m u s t b e particularly careful t o avoid attaching t o o m u c h significance t o inter-pretations of t h e computations if this source of error cannot be remedied.
2 . N U M E R I C A L E X A M P L E
T h e interpretation, in physical terms of t h e experiment, of t h e various p a r t s of a n analysis of variance is t h e final a n d justifying step of t h e entire statistical procedure. Obviously n o general m e t h o d of inter-pretation can be developed t h a t will b e applicable t o all problems. I t will b e considered sufficient t o present t h e analyses of typical investiga-tions, t o serve b o t h as a n indication of t h e power a n d guide for t h e com-p u t a t i o n of t h e statistical com-procedures. One examcom-ple, t h e determination of a n analytical curve, will be presented with full details while t h e others will b e illustrated b y summaries.