Length-Weight Relationships

[Pages:7]Lab. No. 3

Length-Weight Relationships

The significance of Length-weight relationship: (a) Estimation the fish weight based on the known length (b) Measuring changes in robustness /health of this population (relative to past or future samples at the same place and season). (c) Morphometrics interespecific and intrapopulational comparisons.

The relationship between length (L) and body weight (W) for nearly all species of fish can normally be represented by the "length-weight relationship" following equation:

W = qLb

(2.1)

Where W is the body weight of fish (in g), L is the length (in cm) and 'q' and 'b' are constants. The parameter 'b' (also known as the allometry coefficient) has an important biological meaning, indicating the rate of weight gain relative to growth in length or the rate at which weight increases for a given increase in length.

The q and b constants could be estimated from linear functions. However, many functional relationships observed in fishery biology such as length-weight relationship are not linear. Fortunately, such curvilinear functions can often be transformed into linear functions by taking the logarithm or the natural logarithms of both sides:

ln W= ln q+ b.ln L

(2.2)

This equation is equivalent the regression equation:

y = a + b*x

(2.2a)

This mean that; y is equivalent to ln W, a which represents the y-intercept (the point where the line crosses the y axis) of the regression line is equivalent to ln q, b is the slope of the line, and x is equivalent to ln L.

We are now in a position to carry out the estimation of a and b by linear regression analysis.

a = ln q Taking the antilog of a we can calculate q of the original length-weight relationship

q = exp a Note: exp is the inverse of ln, the base of the natural system of logarithms and equal to 2.718282.

For calculation of b -the slope of the line- see the example below.

Thus, the estimated relationship between W (in g) and L (in cm) which is equal to W = q.Lb could be easily determined.

16

Lab. No. 3

Example: The Table below (the shaded cells) represents the Data for estimation of a

length-weight relationship for the threadfin bream (Nemipterus marginatus) from the

South China Sea (from Pauly, 1983). Based on the data in the Table estimate the parameter a and b of a length-weight relationship W = q Lb.

Solution 1. Convert the length measurements to lnL (column no. 4) and the weight measurements to lnW (column no. 5). 2. Square the lnL (column no. 6) and lnW (column 7). 3. Multiply lnL by lnW (column 8). 4. Sum lnL, lnW, (lnL)2 , (lnW)2 ,and (lnL)(lnW) 5. Find the arithmetic mean for lnL and lnW.

i

L

W

1 8.1 6.3

(x) ln L 2.092

2 9.1 9.6

2.208

3 10.2 11.6 2.322

4 11.9 18.5 2.477

5 12.2 26.2 2.501

6 13.8 36.1 2.625

7 14.8 40.1 2.695

8 15.7 47.3 2.754

9 16.6 65.6 2.809

10 17.7 69.4 2.874

11 18.7 76.4 2.929

12 19 82.5 2.944

13 20.6 106.6 3.025

14 21.9 119.8 3.086

15 22.9 169.2 3.131

16 23.5 173.3 3.157

(y) ln W 1.841 2.262 2.451 2.918 3.266 3.586 3.691 3.857 4.184 4.240 4.336 4.413 4.669 4.786 5.131 5.155

(x)2 (ln L)2 4.375895 4.876476 5.393485 6.133242 6.257182 6.888885 7.261016 7.582647 7.892744 8.257374 8.57625 8.669721 9.152386 9.5264 9.804018 9.966652

(y)2

(x)(y)

(ln W)2 (lnL)(lnW)

3.387623 3.85018

5.115572 4.994594

6.007426 5.692184

8.513386 7.225971

10.66518 8.169088

12.8615 9.41283

13.62626 9.946883

14.87267 10.61952

17.50231 11.75335

17.97664 12.18359

18.80075 12.69803

19.47279 12.99322

21.80034 14.12534

22.90411 14.77138

26.3280 16.06612

26.57427 16.27441

n = 16

Total 43.6293 60.78428 Mean 2.726831 3.799018

120.6144 246.4088 170.7767

6. Estimate the slope (b) by means of the relationship

b

=

xy - ( x )( n

x

2

-

(

x

n

)2

y

)

b = 170.7767 -165.7485 = 5.028189 = 3.05735 120.6144 -118.9697 1.644623

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Lab. No. 3

7. Estimate the intercept (a) by means of the relationship a = y -b x

where y and x are the arithmetic means of lnW and lnL respectively a = 3.8 - 8.337 = -4.53786

8. Substitute the a and b values in the regression equation y = a + b*x y = - 4.5379 + 3.0573x

This equation is equivalent to ln W = ln q + b*ln L

Thus, a = ln q = -4.538 we can obtain q of the original length-weight relationship by taking the antilog of a:

q = exp a = 2.718282 (-4.538) = 0.0107

9. Substitute in the allometric equation W = q.Lb to find the relationship between W (in g) and L (in cm) which becomes: W = 0.0107*L3.057

10. Find the Correlation between the length and weight (correlation coefficient)

The correlation between two variables is the degree of association between two variables.

This degree of association is expressed by a single value called a correlation coefficient

(r), which can take values ranging between -1 and +1. When r is negative, it means that one variable (either x or y) tends to decrease as the other increases - there is a "negative correlation" (reveres relationship). When r is positive, on the other hand, it means that the one variable increases

with the other (direct relationship). The correlation coefficient is higher as its value is close to +1 or -1, and is getting

smaller and smaller as it close to zero.

n

xy i - nx y

r=

i =1

n i =1

x

2 i

-

n

x

2

n i =1

y

2 i

-

n

y

2

r=

170.7767 -165.7485

=

5.028189

=

(120.6144 -118.9697)(246.4088 - 230.9206) (1.644623)(15.48825)

r = 5.028189 = 5.028189 = 0.99627 25.47233 5.047233

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Lab. No. 3

Weight (g)

11. Plot the Length-weight relationship of the provided fish species where Y-axis = weight (g), and X-axis = length (cm).

170 160 150 140 130 120 110 100

90 80 70 60 50 40 30 20 10

0 0 2 4 6 8 10 12 14 16 18 20 22 24 26

Length (cm)

Fig. 3.1 Length-weight relationship of Nemipterus marginatus in the South China Sea. (Based on data from the Table)

12. Plot the data after conversion to natural logarithms where: Y-axis = Ln weight (g), and X-axis = Ln length (cm)

Ln (weight)

5.6

5.2

y = 3.0573x - 4.5379

4.8

R2 = 0.9926

4.4

4.0

3.6 3.2

2.8

2.4 2.0

1.6

1.2

0.8 0.4

0.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

Ln (length)

Fig. 3.2 The data from the above Fig. is converted to natural logarithms

19

Lab. No. 3

(Alternative way) Use Excel

Based on the data in the above Table Do the following: a) Plot the Length-weight relationship of the provided fish species where Y-axis = weight (g) X-axis = length (cm)

? Enter the length-weight data on a computer spread sheet (Microsoft Excel)

? Make a scatter plot using the menu choices Insert

Chart

Scatter.

Dialogue box choices permit you to label the axes and title the chart.

? When the diagram is complete, right click on one of the points on the diagram,

select trendline, and choose Polynomial.

b) Plot the data after conversion to natural logarithms where:

Y-axis = Ln weight (g) X-axis = Ln length (cm)

? Convert the length-weight data to natural logarithms. Conduct the ln

transformation of the data by using menu selection

Insert

Function Ln

? Make a scatter plot using the menu choices Insert

Chart

Scatter.

Dialogue box choices permit you to label the axes and title the chart.

? When the diagram is complete, right click on one of the points on the diagram,

select trendline, select Linear and under option check to display the regression

equation of line and the R squared value of the chart.

? Estimate the parameter a and b of a length-weight relationship of the form W = a Lb, usable for predicting weights from length.

Consider the following

Note that, b is close to 3.0 for all species.

? If 'b' is equal to 3.0, growth is isometric, meaning shape does not change as fish grow.

? A value 'b' greater than 3.0 indicates a population where fish become more rotund as length increases.

? A value of 'b' less than 3.0 represents fish becoming less rotund as length increases.

The exact relationship between length and weight differs among species of fish according to:

? Their inherited body shape, and within a species according to the condition (robustness) of individual fish.

? Food availability. ? Sex and gonad development are other important variables in some species.

20

Lab. No. 3

Note: A general rule is that at least 30 fish should be measured to provide a large enough sample size to calculate an accurate regression. Laboratory work Materials

1. Fish, preferably a variety of species; 2. Dissecting equipment (scalpel, forceps, dissecting pins, scissors, etc.); 2. Ruler. 3. Balance 4. Dissecting pan. Procedure 1. Remove one fish from the storage container. 2. Use the available balance to weight the fish individually. 3. Measure the length of fish by means of ruler or on a length board. Length Refers to the whole body length of a fish, there are three types of length measurements: 1. Standard length: is the distance from the tip of mouth to beginning of tail fin 2. Fork length: is the distance from the tip of mouth to the tip of the median caudal fin rays. 3. Total length: is the distance from the tip of mouth to the distal tip of the longest caudal fin ray. Fork Length (FL) is the measurement most commonly used.

For species that do not have a fork in their caudal fins, the Total Length is used instead of fork length.

Differences often exist in the body weight to length relationship for males and females in the same population. If possible, length-weight regressions should be calculated separately for the two sexes (see below).

21

Lab. No. 3

Sex Determination

Determination of sex from external examination of the fish is generally very difficult. For some species, sex may be determined from external secondary sexual characteristics, observable either during the spawning season or, for some species, at any time of year. For most fish species, sex of adult fish can be determined during the spawning season by forcing extrusion of the sexual product (milt (sperm of a male fish)/roe (eggs of a fish)). Still other species have morphological differences which allow determination of sex from external examination at any time. Sex however, could be easily determined for dead fish.

To determine the sex of dead fish follow the following: ? Make an incision on the ventral surface of the body from a point immediately anterior of the anus toward the head to a point immediately posterior to the pelvic fins exposing the gonads. ? If necessary, a second incision may be made on the left side of the fish from the initial point of the first incision toward the dorsal fin. ? Fold back the tissue, to observe the gonads. ? Ovaries appear whitish to greenish to orange and have a granular texture. The eggs will be readily apparent in developed ovaries. ? Testes appear creamy white and have a smooth texture. ? Decide whether the specimen is male or female (testes are solid, ovaries hollow). ? Set out a table as shown below and record the data.

Male

no. L

W

1

2

4

.

Female

no. L

W

1

2

4

.

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