3.1 Data collection
The statistical Catch and Effort Data Recording System (CEDRS) of the Lake Chilwa fisheries has been developed according to the methods described by Bazigos (1972) and has been implemented by Walker (1974). It is based on the method of stratified random sampling. An estimate of total catches (C) is reached through sampling the catch-rates by boat stratified by gear type. The catch-rates (C/f) are raised to total catch by an estimate of total effort by gear (f) obtained through a Frame Survey. Lake Chilwa is stratified into two major strata coinciding with major ecological areas, which are subdivided into five minor strata. Each minor stratum has several beaches on which fish is landed. After the annual frame survey, a number of beaches are randomly selected to record landed catch during so-called Catch Assessment Surveys (CAS). Field-staff spend each month four consecutive days on a beach collecting data according to their monthly CAS itinerary. In total 16 days are spent for fish recording in every month.
During the first day the number and types of boats, fishermen, and type and sizes of fishing gears are counted. In the following three days, all boats leaving for fishing are recorded. Boats landing on a particular beach are recorded and randomly sampled to obtain the fresh weight of the catch by species. The name of the fisherman, amount of fish caught (in kilos), number, size and usage of fishing gear, estimated beach prices and destination of the fish are recorded on the same form. Units of effort are recorded for all seines as number of pulls or hauls; for gillnets it is the number of gillnets of 91 m (100 yards) set. For fish traps the unit of effort is number set and for long lines 100 hooks.
Our analysis of the Lake Chilwa catch and effort data was carried out on four dominant gears although the CEDRS includes in total eight gears. The four gears are fish traps, gillnets, longlines and Matemba seine. A Matemba seine is used either as a beach-seine or is set in shallow open lake areas. It has a length of around 50-300 m, no restriction on mesh sizes, and is operated by five to six persons. Each of these gears represents more than 20 percent of the data and all four in total approximately 92 percent of the data set. Recently, a new important fishery with hooks was included in the CEDRS called nchomanga. Weyl et al. (1999) described this as a passive gear used in densely vegetated areas where it is difficult to find enough space for a long line. A large hook (size 1-3/0) is attached to a length of line onto which a float is attached. This float has enough buoyancy so as not to be submerged by the hooked fish. Alternatively the line is attached to a short length of bamboo which is wedged into reeds or mud to anchor the hook and line. Nchomanga are normally set overnight and baited with small dead fish. Since these data were available for only two years, they were excluded from this study.
Monthly catch per unit effort (CPUE) is estimated as follows (Alimoso Seisay and Zalinge, 1990, FAO 1993): at the end of each survey, all sampled catch-rates are added to obtain a total sampled catch by minor stratum. Similarly, all sampled effort data by gear are added to get a total monthly sampled effort (f) by gear by minor stratum. An estimate of the monthly CPUE by gear is then obtained by through C/f.
The ratio GA, which is termed “gear activity indicator”, is estimated by dividing the total number of gears that were found to be fishing at the time of sampling by the total number known to exist at the landing sites. Total monthly fishing effort, f in the major stratum is estimated by:
f = D* GA* M (1)
where M is the total number of fishing gear units in the major stratum that were counted during a frame survey and D is the number of fishing days in a month The estimated total monthly catch, Y, for a particular gear in the major stratum is:
Y = CPUE × f (2)
Monthly catch and effort data from the Catch Assessment Survey (CAS) for the traditional fisheries sector were obtained from Monkey Bay Fisheries Research Unit and Kachulu Fisheries Office in Zomba in April 1999. The frame survey data were obtained from the Mangochi Fisheries Office and Monkey Bay Research Unit. Water level data, measured daily at the gauge situated near Kachulu, were obtained through the Water Department in Lilongwe.
3.2 Analysis of variance of catch-rates: Differences between years and between months
As processes affecting fish stocks can generally be said to have a multiplicative character, all our analysis are done on ^{10}log-transformed catch-rates: linear trends are thus descriptions of the speed with which these multiplicative processes take place. All analyses were done on mean monthly catch-rate data (CPUE) aggregated over the whole lake. Total CPUE and CPUE by species (group) of four gears fish traps, gillnets, longlines, Matemba seine over a period from 1976 to 1998 were subjected to an Analysis of Variance (ANOVA) with year and month as class variables. This analysis leads to an assessment of the total amount of variation that can be explained by differences between years and between months. The statistical model describing this analysis is:
G(m)_{ijh} = µ_{ijh} + year_{i} + month_{j} + e_{ijh} (3)
Where:
G(m)_{ijh} |
= timeseries of ^{10}log transformed mean monthly catch-rates |
µ |
= overall mean |
Year |
= effect of i^{th} year (1976 - 1998) |
Month |
= effect of j^{th} month (1 - 12) |
e_{ijh} |
= residual error |
3.3 Trend analysis in catch-rates
Trends were analysed with the following polynomial regression model.
G(m) = a + b*year+ c*year*year + e_{t} (4)
Where:
G(m) |
= time series of ^{10}log transformed mean monthly catch-rates |
a |
= intercept |
year |
= represents the linear regression term |
year*year |
= represents the quadratic term |
e_{t} |
= residual error |
To examine whether catch-rates went up, down or remained the same the quadratic part (year*year) of the model was removed and the linear regression was fitted to the monthly data.
Only significant parts of the model were retained, which resulted in four possible models:
1) no regression (when both linear and quadratic terms were non-significant)
2) a linear regression model if only the linear term was significant.
3) a quadratic regression model if only the quadratic term was significant
4) a polynomial regression model if both terms were significant. In this case it was also evaluated how much the quadratic and linear terms each contributed to the explanation of the total variance.
If this amount was very small for the quadratic term, a linear model was chosen
The averages (µ), standard deviations (s) are back-transformed from log-scale to obtain (geometric) means and a factor F around the (geometric) mean. This is done and interpreted as follows:
10^{µ} = 10 to the power of the mean of the log transformed data = geometric mean (GM)
10^{2*s} = 10 to the power 2 x standard deviation This factor (= F) means that that 1 in 20 observations fall outside the range given by F*GM and GM/F.
3.4 Seasonality in catch-rates
Differences between months detected with an Analysis of Variance in (1) give an indication of seasonality. A more formal deterministic description of seasonality could then be made by fitting an appropriate model to the data, and examine the amount of variation explained to judge the strength of the seasonal signal (Zwieten et al., 2002; Densen, 2001). However, as will be seen later, little seasonal effects could be detected in the monthly catch-rate data and this line of inquiry was not continued. As the observed variation between months in the data therefore is not predictable it becomes part of the basic uncertainty in the data.
3.5 Basic uncertainty in catch-rates
The total amount of variation explained through the year effect in the statistical model (3) expresses differences between years. This includes both short-term annual variation and trends and long-term trends (long-term = over the whole series examined). Annual variation, including short-term trends, and seasonality obscure the long-term trend. The total amount of residual variation after removing a long-term trend and seasonality can be deemed basic uncertainty. This part of the total variance that cannot be explained by analysis of trends and seasonality consists of process error (i.e. natural variation), measurement error and observation error. If no variability can be explained by trend and seasonality then the total variation is basic uncertainty.
3.6 Trend-to-noise: power analysis
The slope parameter b in the linear regression (2) divided by the standard deviation(s) of the residuals of the linear regression, is the trend-to-noise ratio. How the trend-to-noise ratio affects the number of years (n) over which a trend may be perceived with a power (1-?), given probabilities for the statistical decision levels of a type I error? and a type II error? can be derived as follows (Zwieten et al., 2002; Densen, 2001).
An estimate of the variance of the slope estimate b is:
(5) |
where s^{2 }is an estimate of the variance in the residuals around the regression line, t is the independent variable time and n is the number of observations. If catch-rate estimates are taken at regular intervals in time (or space), s_{t}^{2} ca n be rewritten as (Gerodette, 1987):
(6) |
The power of a test is the probability that a decision rule will lead to the conclusion that an alternative hypothesis H_{a}: b_{0}¹f is true, i.e. that a trend or deviation from a trend will be detected in the cases of f = 0 or some specified value. The test statistic for b is t*= (b-f)/s_{b}. This probability (P) is given by:
(7) |
where d is a measure of non-centrality, or how far the true value b_{0} of b is from H_{0}: b_{0} = f:
(8) |
To reduce the statistical errors to the specified levels of? and? = 0.05 or 0.1 as used here, with the zero hypothesis of no trend (H_{0}: b_{0} = f = 0) the following inequality should hold:
(9) |
for a two tailed test. Substituting (5) and (6) into (9) gives:
(10) |
In the presence of auto-correlation the variance the residuals is underestimated by a factor 1/(1-r^{2}), where r is an estimate of the auto-correlation coefficient r (Neter et al., 1985; Gerodette, 1987). We studied the effect of serial correlation at lag 1 on trend perception by including this factor in equation 7, leading to:
(11) |
where:
b |
= trend parameter (slope) in the linear regression |
n |
= the number of observations |
t_{a/2}^{,} t_{b} |
= the test statistic or decision rule of a t-distribution, where a is the specified probability of making a type I error (a trend is rejected where in fact there is a trend) and b the specified probability of making a type II error (a trend is accepted where in fact there is none). In our case a = b = 0.1 or both errors are set at the 10 percent level. |
s |
= the standard deviation of the residuals |
This formula is solved for n (the number of months of data collected) with given trend and variance. From the formula it can be inferred that the variance in a series exhibiting auto-correlation will be reduced, which could result in a conclusion of a trend where in fact there is no trend.
3.7 Analysis of water levels related to catch-rates
^{10}Log-transformed annual average catch-rate series were de-trended by subtracting the linear trend from the series through linear regression. There was no need to de-trend the water level series as there was no long-term trend present in the time series and the seasonal signal in monthly variation was low. The resulting residuals of the catch-rates were subsequently cross-correlated with annual mean, minimum and maximum water levels. Cross-correlation is done by shifting the two series with steps of one year against each other and calculate correlations at each successive lag. As the two series were 20 years with a maximum of 18 data up to five lags could be investigated. This, however, is sufficient as the regenerative response (recruitment processes) of the species under investigation to environmental changes is fast -for Barbus probably even within a year (see earlier and Kalk, McLachlan and Howard-Williams, 1979). Subsequently, the lags with highest correlation coefficients were investigated through regression analysis. The amount of variability explained by the regression analysis is an indication of the magnitude of the effect of changing water level that can be seen in the time series. A similar analysis was done on monthly catch-rates correlated with monthly water levels, both after de-trending. This analysis did not yield more information than the previous analysis and is therefore not presented.
3.8 Multiple regression of water levels and fishing effort on catch-rates
Multiple regressions were performed with ^{10}log-transformend total catch-rates by gear, or catch-rates by species(group) by gear, as dependent variable. Fishing effort as total number of gears and annual mean, maximum or minimum water levels - the latter with or without a lag phase from the cross-correlation analysis - were the explanatory variables. Both water levels and fishing effort each are made orthogonal by subtracting the mean from the original series. In the previous analysis it was established whether there was a lag-phase between de-trended annual mean water levels and annual mean catch-rates, and which of minimum, maximum or mean water levels was more informative. Missing values for numbers of gear, as taken from frame surveys, were interpolated by linear regression. Lag(1) means that previous years water level is compared with this years catch-rate.
The multiple regressions were always of the form:
G(m)_{t} = µ + effort_{t} + water level_{t-lag(x)} + effort_{t}*water level_{t-lag(x)} + e_{t} (12)
Where:
G(m)_{t} |
= time series of annual mean 10log(CPUE) by gear (1979-1998) |
µ |
= overal mean |
effort_{t} |
= total number of gear |
water level_{t-lag(x)} |
= water level at lag(x) where x = 0 - 5 |
effort_{t}*water levelt_{t-lag(x)} |
= interaction of effort and water level at lag(x) |
e_{t} |
= residual error |
In all cases non-significant explanatory variables were removed from the model. The interaction effect is interpreted as reflecting possible changes in catch-rate as a result of changes in efficiency or of usage of gears in relation to water levels. Such changes could be a result of spatial effects of accessibility of species to gears and the effectiveness of gears (e.g. concentration of fish with receding water levels). All ANOVA and regressions are carried out with the General Linear Models procedure (SAS Institute Inc., 1993). Cross-correlations are carried out with the ARIMA procedure (SAS Institute Inc., 1989).
3.9 Analysis of effort data
All effort data were compiled by major strata, and graphically displayed with regression lines to display trends. Total effort data of Lake Chilwa are considered unreliable, in particular concerning fishing operations conducted by migrant fishermen in the swamp area of lake who live in temporary shelters (Zimbowera). These fishermen are not recorded. Fish landed on beaches from these operations is mostly in dried form ready for marketing. The swamp areas are difficult to access. However, as all annual frame surveys are conducted in a similar fashion, the effort data can be considered indicative for relative changes taking place.