SAMPLE SIZE

Unlike other types of sampling, this problem involves living plankton in water. The unit for the sample is the water drawn from the source. The plankton is found in the water. The unit of sample is one liter of water with whatever count of plankton inside that one litter. This is a population density issue: number of plankton per liter of water. Minimum sample size is not critical for this experimental design.

TEST FOR REGULARITY

Assume that series of samples will be taken from the water for plankton count, i.e. 2 samples per month are taken from the water over a period of one year. There will be 24 samples taken in toto. Assume further that these 24 samples are bootstrapped in approach. Each sample drawn is 1 litter of water. The plankton count is plankton count per liter. With these two samples drawn, it is possible to determine whether the population of the plankton in the water changes by computing paired means (d-bar analysis). There is a minimum of 12 pairs in 24 month and the maximum pairs are determined by:

(1) Pairs max = ([n(n - 1) -2]/2) + 1

A series of 24 samples would produce a total of 300 possible pairs. However, it may not be necessary to run 300 d-bar analysis. It is adequate to run 12 d pairs to find the paired difference mean (d-bar) which is given by:

Paired difference:

d1 - d13 = d*1

d2 - d14 = d*2

...

d12 - d24 = d*12

Paired difference mean:

(2) d^ = 1/n(sum d*)

The test statistic is given by:

(3) td = d^/(S/sqrt n)

... where td is the critical Student t-value. At 0.95 standard of review at degree of freedom = 24 -1 23 is 2.069. If use df infinity, t-critical is 1.64. S = standard deviation among d* series: {d*1, d*2, ...,d*12} and n = 12 pairs. If the observed td is larger than 2.069, then there is a significant different in plankton counts among 24 months. In that case, the maximum and minimum number be be determined. Graphically, this can be done by plotting d1, d2, ...,d24. A peak and trough would be found in the series.

ANDERSON-DARLING TEST TO CONFIRM MAXIMUM AND MINIMUM POINTS

The point where the number of plankton is maximized and minimized may be found by using the Anderson-Darling test. Since there are 24 sample counts, rank the samples in series of time: t1, t2, ...,t24. Break these samples into 4 blocks of 3 months or 6 samples each. The Anderson-Darling test verifies whether the data produces normal distribution. If A^2 < A*2 then normal; if A^2 > A*2 then not normal where A^2 is the observed value and A*2 is the standard value set at 0.787 for unknown population mean and unknown population variance.

SLOPE TEST

Another method is to take the 4 blocks describe above and run a time series regression:

BLOCK 1 (x1: t1), (x2: t2), (x3: t3)

BLOCK 2 (x4: t4), (x5: t5), (x6: t6)

BLOCK 3 (x7: t7), (x8: t8), (x9: t9)

BLOCK 4 (x10: t10), (x11: t11), (x12: t12)

Run regression for each block and compare the slope. If the sign of the slope changes, it means that an inflection point occurs in that block. A minimum and maximum points may be determined in this manner.

MARKOV CHAIN UNADVISABLE

Markov chain by definition is the forecast of the next event by using a series of events immediate preceding the current estimate time. This is not a good method for estimating the current population on the basis of the prior value since we are dealing with uncontrollable and unpredictable drop out rate, i.e. death of plankton and environment changes or changes in water condition that may affect survival rate of the plankton.

POPULATION ESTIMATE

One means to estimate the general population is the use of tag-release-recapture method. The method may work well with large animals; however, with microscopic plankton, it might not be feasible. It is best to estimate the density of the distribution of plankton counts per cubic meter of water in the location of study.

There may be an alternative means to estimate the total population. use the Weibull Distribution by defining the number of plankton captured in each sample as "age to failure" or surviving up to that time. Assume that two samples are taken each month. There will be a total of 24 samples in one year. The sample series would be: {n1, n2, ...,n24} with the corresponding time series: {t1, t2, ...,t24}. The plankton counts would be correspondingly: {x1, x2, ...,x24}. The following procedures are used:

First, find the Y-plot position called F(t) where F(t) is given by:

(4) F(t) = (i - 0.3)/(n +0.4)

... where i is the individual sample. There are a total of 24 samples: n = 24 in the series for time {t1, t2, ...,t24} months respectively.

In order to run a Weilbull forecast, it is necessary to generate the X and Y value for the linear regression equation in a form of Y = a + bX + c. The definition for X and Y follows:

(5) X = ln(ln(1/(1-F(t))) , and

(6) Y = ln(x)

Use the result in (2) and (3) to run a regression analysis. An forecast equation in a form of Y = a + bX + c should be obtained and from which the Weibull statistics are determined. The shape of the Weibull distribution follows:

(7) Beta = 1/b

... where b is the slope from the Weibull regression line obtain from equations (2) and (3). The characteristic of Weibull beta has the following property:

Beta 1 means the number of plankton is decreasing with time (dying)

The current death or instantaneous failure rate is given by the Weibull hazard function:

(8) H(t) = (beta/Eta)(x/eta)^beta-1

If you have 24 counts of samples, you should have 24 H(t) and can use it to run a forecast of plankton death rate (attrition).

The Weibull intercept called eta (Greek letter) is given by:

(9) Eta = exp(a)

... where a = y intercept of the Weibull generated linear regression line Y = a + bX + c.

The Weibull reliability is given by:

(10) R = 1 - CDF

This means that at any time, you should expect to find this amount of plankton count per cubic liter of water drawn from the target location, i.e. sea or bay.

STOCHASTIC DRIFT & SEASONAL FLUX

The bootstrap sampling process suggested above would be able to generate series of mean set, i.e. mean t1, t2, ... tn for comparison. if there is a difference among the means series, then one may reasonably conclude that there is a stochastic drift among the general population. If there is a change in water condition, this stochastic drift should be expected. The stochastic drift is an evidence of seasonal flux. For hypothesis test and confidence interval procedure, see d-bar analysis described above.

SEASONAL FLUX AND FORECAST EQUATIONS

The 4 blocks analysis above will prove whether there is population flux. If there is a flux (also evidenced through stochastic drift), the forecast model must be adjusted for the seasonality. It is recommended to construct a forecast model for each block that is specific for a certain time of the year (seasonal). In order to stabilize the forecast model, data from more than one year is required to see repetition of patterns.

STATIONARY NOT RECOMMENDED

Unless the animal is in a laboratory, it is not advisable to use this approach. In bacteriology, for instance, where bacteria is grown in petri dish and fed with specified food and kept under a certain environmental condition, it is predictable that the growth would follow the conventional logistic function pattern of the S-curve. Since all bacteria will follow this growth pattern, it may be assumed that the condition is stationary. However, for plankton in the open water---and since plankton density in water may be used to indicate environmental condition or water quality, the assumption of stationary would mislead the researcher to overlook plankton count outside of the best fit region of the S-curve. it is better not to assume a population pattern, but to verify the population pattern through empirical data.

Thanks Paul, this is very helpful. If I understand it correctly, the d-bar analysis only works if we have at least 2 yearly cycles. However, let us assume we only have one yearly cycle available. The data we have are flux data, collected from a sediment trap. Thus it provides an estimate of the total population size dropping dead during a certain period. typically, the sampling is continuous over the year, but may have irregular intervals. It appears to me that we have three theoretical situations:

a) the flux is uniform

This would indicate that stochastic noise and population dynamics occur at a time scale much shorter than that of the sampling interval.

b) the flux is random

This would mean the flux may drift, but not beyond a red noise. I would not discard red noise and Markov chain so fast, because the development of the flux of a population surely must be to a certain degree an autocorrelated process. You cannot generate an arbitrary number of offspring.

c) the flux shows a seasonal pattern

When none of the above are true then the flux shows a pattern indicative of a significant deviation from uniformity and randomness - the flux changes and it changes more than what could be expected by chance.

The question at hand is how to distinguish these three situation, when only one yearly series is present. I.e. where we cannot carry out a recurrence analysis of any kind.

DATA PARTITION

Where the data is collected from series of samples: k samples, over a period of one year, this data set can be partition into its individual collection. Treat each collection as a separate sampling, thus: {x1, x2, ..., xn}. What we have on hand then is a set of bootstrapped samples. Let us follow through the three scenarios presented on population flux.

EVIDENCE OF FLUX

in order to undertake testing for any one of the three or all of the three scenarios for flux, the existence for flux must be proved. Each sample took within the year, i.e. each bootstrapped sample may be treated as a data set for regression analysis. Regression may be run in two ways: (i) regression of plankton count per time, i.e. (x1:t1), (x2:t2), ..., (xn:tn), or (ii) regression in one block for the entire year.

(i) Regression per sample

This type of regression may be accomplished through the Weibull process. Since Weibull result communicates information about failure (death), the survival number of the plankton then is 1 - F(t). The regression equation for each sample will show its own slope. Assume that there were 12 samples taken in one year (one sample per month) then there will be 12 Weibull Y = a +bX + c. Take the 12 slopes and plot them as one series, one would have a pattern.

A second option under this scenario is to pick 3 collections: (x1:t1), (x2:t2) and (x3:t3) and run a time dependent regression. Keep doing it for the rest of thesamples in the year and monitor slope change. If there is a changes in slope, it means there is a flux. If the slope changes from positive to negative, it means that the growth had maximize at that interval: (x1, x2, x3). If the slope change from positive to negative and then from positive back to negative again, there is a repeating pattern. To test whether the flux is seasonal or random is to do b-bar analysis for the differences of the slope.

(ii) Regression of the data for the entire year as one block (set)

Only one regression equation will be obtained. this is not advisable. No pattern of fluctuation may be observed.

UNIFORM FLUX

Uniform flux means that the change is predictable and that the rate of change is uniform. For example, the log function manifesting the S-curve for bacterial growth is uniformly predictable. However,

RANDOM FLUX

Use (i) above, if the slope comparison according to signs: compare positive slope with positive slope, and negative slope with negative slope. If the (+:+) shows significant difference then there is a random flux on growth and (-:-) slope comparison on plankton death rate.

SEASONAL PATTERN FLUX

One would see a repetitive pattern of the slope (same sign) of various interval carried out in (i) and these variations are not statistically significant. See r in 91) and (2).

POPULATION GROWTH MODEL

In general, a population model is given by:

(1) dP/dt = rP(1 - P/K)

... where r = growth rate, P = population size, and K = carrying capacity. The growth rate may be taken as the slope from the regression run above.

Assume that the first sample at t1 is used as the initial point, the population growth monitoring model may be given as:

(2) P(t) = KP'(exp (rt)) / K + P'(exp(rt) - 1)

... where P' = initial population count; and t = time.

With the slope known from the regression runs and P initial known from the first sample, one can determine the carrying capacity (K) and monitor the changes in P in subsequent period. If there is a change in P in subsequent periods, that may mean (i) the rate of growth change, i.e. slope of regression line change, or (ii) K change---if K changes, it may mean that something is changing in the water to make the plankton die or grow faster.

ONE COLLECTION

If in one year, only one collection is taken from a sediment, no pattern may be determined. This is not a data set; this is a point in space. Following the logic of two points make a straight line. A line makes a pattern. A pattern presents a forecast model.

I tend to prefer regression because it is simple, direct, robust, and gives a readily understandable and graphable model. Your question says “event” but given that you are speaking of plankton, which presumably you have sampled, you are looking for some feature in the series of plankton data.

Given that you say “cyclic nature ... the end of the time series is linked to its beginning”, you have recognised that you must use a method that can accommodate that.

If your dependent variable Y can be treated as continuous variable, and large integers like plankton counts usually can be, the obvious choice would be PERIODIC REGRESSION.

I would recommend you construct a periodic regression model that will let you address multiple cycles simultaneously, because you are very likely to find that the second harmonic of the year is useful in addition to the first harmonic (365 d cycle), and likewise the time of day, and not unlikely the lunar phase, and possibly also the tide.

There is no problem using data from multiple years; but when doing so, instead of discarding the year identity, construct a variable to represent sequential days since the beginning of the study (you could use years instead [i.e. Yr1, Yr2, ..., Yrn] but that would be assuming a step-like change and assuming that the step occurs at Jan 0, but that assumption would require justification and if wrong could affect the analysis) to accommodate a continuous effect. Continuous effects could represent small sections of longer cycles that may remain unspecified (like sunspot cycles) or could simply represent drift in calibration or drift in the performance of your nets. With periodic regression there is no problem also including quality-control variables to identify things like what equipment was used, or what team was responsible for what samples, etc. The number of explanatory variables should be small relative to the number of data.

Your model thus might be something like

Y = B0 + B1*sin`DOY + B2*cos`DOY + ... + Bn*NDOY + error

... where DOY is Day of Year (0-364); NDOY is the number of days since any date prior to the study (actually it can be since a date after the study, but let’s keep it simple); the grave (`) symbol means “proper” and you would read sin`DOY as “the proper sine of Day of Year”, “sine of time in year”, etc. (this notation merely makes it much easier to talk about what the intent is rather than get bogged down in details that are not conceptual, e.g. ‘oh, you are dealing in radians, so we write B1*sin(DOY*2pi/365)’. The notation means you are doing whatever is proper to take the sine of the time in the year. Finally, of course B0,1,2, ... n are the coefficients you wish to determine. You will notice that periodic regression is unusual in that the Y need not be transformed, but the X (DOY, or time of day, or tide, or lunar month) is, and you don’t take just one transform, you take two, because sine and cosine are the fundamental descriptors of an angle. It is clear that you can add pairs of terms for any other cycle like time of day (ToD), e.g. “...+ B3*sin`ToD + B4*cos`ToD”. You can also add dummy variables for qualitative factors like data source.

See:

Bell, K.N.I. 2008. Analysing Cycles in Biology and Medicine—a practical introduction to circular variables & periodic regression. 2nd edition, Razorbill Press. ISBN 978-0-9736209-2-4. xv+163 pp.

(You can view much of the book’s contents on Amazon. The aim of “Analysing Cycles...” is to make this under-known technique accessible. Periodic regression itself is discussed in Ch 6; prior chapters explain what you need to know about circular variables so that Ch 6 will be followed. The book is not designed to be read cover-to-cover, so by all means begin with Ch 6.)

And see examples in (downloadable from my publications page ucs.mun.ca/~kbell/pubs) papers:

Bell K.N.I. 2007. Opportunities in Stream Drift: Methods, Goby Larval Types, Temporal Cycles, In situ Mortality Estimation, and Conservation Implications. In Biology of Hawaiian Streams and Estuaries. Edited by N.L. Evenhuis & J.M. Fitzsimons. Bishop Museum Bulletin in Cultural and Environmental Studies 3: 35–61.

Bell, K. N. I., Pepin, P., & Brown, J. A. 1995. Seasonal, inverse cycling of length- and age-at-recruitment in the diadromous gobies Sicydium punctatum and Sicydium antillarum (Pisces) in Dominica, West Indies. Can. J. Fish. Aquat. Sci. 52:1535-1545.

Note that I advocate that sampling be, over time, haphazard or random. If your sampling is regular, you have been conventional and you are probably okay, but there is a possibility that, because you are in effect introducing an artificial periodicity and depending how this interacts with real periodicities in the system being sampled, there is the possibility of effects, not unlike Moiré effects, emerging and seeming real when they are artefactual. That is not a big worry if you stick to the main astronomically-related cycles (time in year, in lunar month, in day, and in tidal cycle) but could be an issue if you were exploring an infinity of cycles. There would be no point in, for example, analysing for any cycle, e.g. an 11-day or 17-day, for which there is not a ready astronomical connection and a reason a priori to expect an effect. Also, do not waste your time trying to test for cycles other than those for which there is plausible theoretical biological response; i.e. do not waste your time trying to say, e.g., “is this a yearly cycle or is it a 357-day cycle”. Stick with the key astronomical cycles (year, lunar month, day, tide) and (sometimes) their harmonics.

I hope this helps. You are welcome to contact me directly (kbellATmunDOTca) if you have other questions.