Analysis of variance and statistical inference Plot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Light intensity Biomass A 10 A 11 A 7 A 9 A 9 B 4 B 6 B 3 B 8 B 5 B 9 B 5 C 15 C 10 C 4 C 11 C 6 C 9 D 9 D 5 D 7 D 3 D 5 D 6 D 7 Plot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Light A A A A A B B B B B B B C C C C C C D D D D D D D Biomass 10 11 7 9 9 4 6 3 8 5 9 5 15 10 4 11 6 9 9 5 7 3 5 6 7 } } } } i 1 2 A } B 2 C 2 D 2 ni 2 ( xi , j xtotal ) i 1 j 1 k SStotal k SSbetween ni ( xi xtotal ) 2 ni 2 ( xi , j xi ) i 1 j 1 k 2 SS within Plot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Light A A A A A B B B B B B B C C C C C C D D D D D D D Biomass 10 11 7 9 9 4 6 3 8 5 9 5 15 10 4 11 6 9 9 5 7 3 5 6 7 } } } } 2 2 A SStotal SSbetween SSwithin B dftotal dfbetween df within } 2 C 2 D 2 2 between F 2 within SSbetween N k SSbetween F k 1 SS within k 1 SS within N k Observations 1 2 3 4 5 Group mean SSwithin Total SSwithin Total SSbetween Grand mean Grand SS Grand SS SSbetween+SSwithin F F-test A 0.08 0.71 0.19 0.51 0.73 0.445 0.131 0.070 0.065 0.004 0.082 4.11 13.96 1.08 1.00 0.14 0.79 0.32 0.12 18.07 18.07 18.14 2.118E-05 Treatments B C 0.19 0.83 1.21 0.71 1.97 1.10 0.19 0.11 0.19 0.30 0.750 0.611 0.319 0.046 0.216 0.010 1.484 0.244 0.314 0.250 0.312 0.096 SSbetween D 2.80 2.69 1.93 2.57 2.58 2.515 0.081 0.032 0.342 0.004 0.004 0.404 0.404 0.404 0.404 0.404 0.109 0.109 0.109 0.109 0.109 0.220 0.220 0.220 0.220 0.220 k SSbetween ni ( xi xtotal )2 i 1 ni 2 ( xi , j xi ) i 1 j 1 k 0.80 0.02 0.79 0.79 0.79 0.06 0.14 0.00 0.94 0.61 2.96 2.61 0.72 2.23 2.24 2.059 2.059 2.059 2.059 2.059 SS within ni ( xi , j xtotal )2 i 1 j 1 k SStotal Repetitive designs In medical research we test patients before and after medical treatment to infer the influence of the therapy. We have to divide the total variance (SStotal) in a part that contains the variance between patients (SSbetween) and within the patient (SSwithin). The latter can be divided in a part that comes from the treatment (SStreat) and the error (SSerror) Medi cal treat ment k SStotal SSbetween SSwithin SStreat SSError SStotal SSbetween SSwithin SSbetween SStreat SSerror n SStotal (x ij x) 2 j1 i 1 df total df between df within df between df treat df error n SSbetwen k (Pi x) 2 kn 1 n 1 n(k 1) n 1 k 1 (n 1)(k 1) i 1 k n SSwithin (x ij Pi ) 2 j1 i 1 k n (Tj x) 2 k SStreat n (Ti x) 2 F j1 k n SSerror (x ij Pi Tj x) j1 i 1 2 SStreat df error SSerror df treat j1 k n (x j1 i 1 ij Pi Tj x) 2 (n 1)(k 1) k 1 Person 1 2 3 4 5 6 7 8 9 10 Mean within Grand mean SSbetween df Morning 104 104 84 97 81 108 89 82 84 95 92.8 Afternoon 119 96 114 114 111 116 96 96 129 120 111.0 98.7 Squares SSwithin df 20.2 29.1 196.8 18.5 203.8 15.4 12.1 30.4 310.3 21.1 857.6 106.8 3.5 246.3 168.6 230.4 145.9 9.9 69.3 747.1 410.7 2138.4 34.2 12.4 2.8 75.3 0.8 255.9 0.1 7.9 94.5 245.7 729.6 34.8 152.4 41.6 n=10 SStreat df Squares SSerror df F p(F) Night 102 95 96 92 95 88 93 85 91 84 92.2 2.0 127.5 66.1 2.5 70.2 96.4 5.8 0.1 137.3 1.7 Sum 1437.6 18.0 14.324 0.067 4.0 202.3 11.2 0.4 8.0 0.1 84.6 16.2 224.6 62.7 Mean persons 108.2 98.3 98.0 101.1 95.5 104.3 92.6 87.9 101.2 99.7 Squares 90.2 0.1 0.4 5.9 10.2 31.6 36.5 116.6 6.4 1.0 sum k=3 299.0 897.0 Mean within Grand mean SSbetween df Squares Sum 3725.6 20.0 228.8 2288.0 2.0 0.4 8.6 22.9 5.0 30.7 91.2 46.0 13.2 10.7 85.2 Grand SS SSwithin+SSbetween SSwithin SStreat+SSerror Person 1 2 3 4 5 6 7 8 9 10 Afternoon 10 -2 16 13 15 12 3 8 27 20 12.3 Ipsative data Night -6 -4 -2 -9 -1 -16 0 -3 -10 -16 -6.4 Mean persons 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Squares 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 sum k=3 0.0 0.0 0.0 k n 2 SStotal 20.2 (x ij x) 106.8 34.2 j1 i 1 3.5 29.1 12.4 196.8 246.3 2.8 n 18.5 168.6 75.3 2 203.8 230.4 0.8 betwen i treat error 15.4 145.9 255.9 i 1 k 0.1 n 12.1 9.9 error treat n69.3 30.4 k 7.9 310.3 747.1 94.5 2 i i1 within 21.1 410.7 ijj1 245.7 1 857.6j1 i2138.4 729.6 k n(n 1) Tj SSSS df k (P x) j1 F SS df (x ij Tj ) 2 SS (x P ) Sum SSwithin df SStreat df Squares 34.8 k SStreat n (Ti x) SSerror df F p(F) 152.4 41.6 n=10 2 j1 2.0 k n 4.0 127.5 202.3 11.2 ij error 66.1 2.5j1 i 1 0.4 70.2 8.0 96.4 0.1 5.8 84.6 0.1 16.2 137.3 224.6 1.7 62.7 SS 4622.6 4622.6 3725.6 3725.6 Morning -4 5 -14 -4 -14 4 -3 -6 -18 -5 -5.9 Sum 2 3725.6 20.0 228.8 2288.0 2.0 0.4 8.6 22.9 i 5.0 30.7 91.2 46.0 13.2 10.7 85.2 (x P Tj x) 2 1437.6 18.0 14.324 0.067 Grand SS SSwithin+SSbetween SSwithin SStreat+SSerror 3725.6 3725.6 3725.6 3725.6 Spiders from two Mazuarian lake ensembles Summary statistics Islands IslandEns emble Area GórnaE MNB 0.7 Koń MNB 0.5 Kopanka MNB 0.69 KrólewskiOstrów MNB 6.15 Maleńka MNB 0.0003 MałaWierzba MNB 0.4 KopankaN MNB 0.18 Ośrodek MNB 0.09 Piaseczna MNB 0.63 Ruciane MNB 10 Mikołajki MNB 10 Śluza MNB 0.48 GórnaW MNB 0.44 Wierzba MNB 0.78 Wygryńska MNB 0.67 Bryzgiel Wigry 0.2 Bryzgiel Wigry 10 BrzozowaL Wigry 3.81 BrzozowaP Wigry 2.32 CimochowskiGrądzikC Wigry 0.15 CimochowskiGrądzikN Wigry 0.14 CimochowskiGrądzikS Wigry 0.76 Kamień Wigry 3.13 Krowa Wigry 4.49 MysiaWigry Wigry 1.55 Ordów Wigry 8.69 Ostrów Wigry 38.82 Rośków Wigry 0.56 Walędziak Wigry 0.76 WysokiWęgiełWigry 10 Distfromn earestMai nland Latitude 200 161 131 123 40 180 237 5 290 0 0 30 287 160 120 30 0 220 180 40 170 70 170 120 60 140 190 100 30 0 53.63397 53.62628 53.62765 53.63025 53.64253 53.76169 53.63135 53.63747 53.68015 53.62833 53.7854 53.66271 53.6343 53.7592 53.68694 54.00219 54.00886 54.02619 54.02658 54.05194 54.05203 54.04875 54.02625 54.01289 54.07183 54.00739 54.00636 54.00217 54.00344 54.03497 Organ Temp SoilH SoilD SoilF SoilA icMatt Distur Speci Individuals Longitude Traps Light eratur umidit disper Individ ertility cidity eCont bance es /trap e y sion ent 21.54726 21.51939 21.54238 21.54168 21.56479 21.60678 21.54744 21.54389 21.56987 21.52552 21.58318 21.5731 21.54433 21.6061 21.56201 23.07553 23.09219 23.10886 23.12553 23.07553 23.07553 23.07553 23.12553 23.09219 23.09219 23.05886 23.07553 23.07553 23.05886 23.12553 4 4 4 12 1 4 3 2 11 5 15 4 5 8 5 3 6 3 3 3 3 3 11 6 6 10 15 3 3 10 3.04 3.15 3.37 3.23 3.84 3.74 3.55 3.47 3.49 3.06 3.23 3.36 3.13 3.24 3.61 3.77 3.52 3.62 3.68 3.57 3.55 3.61 3.75 3.72 3.76 3.71 3.68 3.66 3.70 3.45 3.67 3.54 3.52 3.66 3.52 3.61 3.67 3.54 3.65 3.55 3.49 3.51 3.56 3.57 3.48 3.70 3.51 3.59 3.58 3.55 3.52 3.65 3.62 3.57 3.63 3.64 3.55 3.65 3.56 3.57 3.20 3.41 3.24 3.24 4.55 4.76 3.08 4.11 3.33 3.37 3.44 3.92 3.24 3.55 3.54 4.49 3.58 4.47 4.57 4.36 4.63 4.20 3.26 4.64 3.85 3.15 3.33 4.24 4.41 3.59 3.25 3.89 3.11 3.68 3.77 3.66 3.27 3.81 3.98 3.77 3.70 3.82 3.48 3.89 3.91 3.36 3.37 3.33 3.53 3.51 3.66 3.51 3.54 3.36 3.70 3.42 3.53 3.24 3.49 3.44 3.78 4.01 3.50 3.94 4.11 4.05 3.74 4.04 4.16 3.96 3.93 4.05 3.94 4.10 4.10 3.75 3.76 3.63 3.79 3.80 3.89 3.89 4.03 3.72 4.00 3.99 3.96 3.63 3.81 3.86 3.92 3.93 3.82 3.99 4.04 4.35 3.82 3.89 3.76 3.83 3.90 4.00 3.79 3.80 3.67 3.65 3.74 3.51 3.76 3.73 3.86 3.79 3.82 3.56 3.82 3.82 3.83 3.72 3.85 3.72 2.11 2.08 2.08 2.00 2.49 2.58 2.02 2.38 2.06 2.05 2.10 2.19 2.06 2.10 2.19 2.71 2.24 2.75 2.69 2.60 2.70 2.50 2.09 2.79 2.33 2.08 2.17 2.51 2.59 2.27 1 4 1 2 3 4 1 4 1 3 3 4 1 3 2 4 4 4 4 4 4 4 3 4 3 3 4 4 4 4 33 25 34 51 6 27 32 22 38 28 75 19 29 47 43 21 46 31 30 25 31 25 60 34 49 64 93 24 28 57 149 275 349 920 12 83 92 128 616 176 673 281 204 687 912 124 244 360 232 188 258 170 440 347 386 587 914 154 88 307 37.25 68.75 87.25 76.66667 12 20.75 30.66667 64 56 35.2 44.86667 70.25 40.8 85.875 182.4 41.33333 40.66667 120 77.33333 62.66667 86 56.66667 40 57.83333 64.33333 58.7 60.93333 51.33333 29.33333 30.7 Starting hyotheses • The degree of disturbance (human impact) influences species richenss. • Species richness and abundance depends on island area and environmental afctors. • Island ensembles differ in species richness and abundance. • Area, abundance, and species richness are non-linearly related. • Latitude and longitude do not influence species richness. Sorting • Area, abundance, and species richness are nonlinearly related. • Latitude and longitude do not influence species richness. • Species richness and abundance depends on island area and environmental factors. • Island ensembles differ in species richness and abundance. • The degree of disturbance (human impact) influences species richenss. The hypotheses are not independent. Each hypothesis influences the way how to treat the next. • Area, abundance, and species richness are non-linearly related. Species – area and individuals area relationships Islands IslandEns emble Area GórnaE MNB 0.7 Koń MNB 0.5 Kopanka MNB 0.69 KrólewskiOstrów MNB 6.15 Maleńka MNB 0.0003 MałaWierzba MNB 0.4 KopankaN MNB 0.18 Ośrodek MNB 0.09 Piaseczna MNB 0.63 Ruciane MNB 10 Mikołajki MNB 10 Śluza MNB 0.48 GórnaW MNB 0.44 Wierzba MNB 0.78 Wygryńska MNB 0.67 Bryzgiel Wigry 0.2 Bryzgiel Wigry 10 BrzozowaL Wigry 3.81 BrzozowaP Wigry 2.32 CimochowskiGrądzikC Wigry 0.15 CimochowskiGrądzikN Wigry 0.14 CimochowskiGrądzikS Wigry 0.76 Kamień Wigry 3.13 Krowa Wigry 4.49 MysiaWigry Wigry 1.55 Ordów Wigry 8.69 Ostrów Wigry 38.82 Rośków Wigry 0.56 Walędziak Wigry 0.76 WysokiWęgiełWigry 10 Distfromn earestMai nland Latitude 200 161 131 123 40 180 237 5 290 0 0 30 287 160 120 30 0 220 180 40 170 70 170 120 60 140 190 100 30 0 53.63397 53.62628 53.62765 53.63025 53.64253 53.76169 53.63135 53.63747 53.68015 53.62833 53.7854 53.66271 53.6343 53.7592 53.68694 54.00219 54.00886 54.02619 54.02658 54.05194 54.05203 54.04875 54.02625 54.01289 54.07183 54.00739 54.00636 54.00217 54.00344 54.03497 Organ Temp SoilH SoilD SoilF SoilA icMatt Distur Speci Individuals Longitude Traps Light eratur umidit disper Individ ertility cidity eCont bance es /trap e y sion ent 21.54726 21.51939 21.54238 21.54168 21.56479 21.60678 21.54744 21.54389 21.56987 21.52552 21.58318 21.5731 21.54433 21.6061 21.56201 23.07553 23.09219 23.10886 23.12553 23.07553 23.07553 23.07553 23.12553 23.09219 23.09219 23.05886 23.07553 23.07553 23.05886 23.12553 4 4 4 12 1 4 3 2 11 5 15 4 5 8 5 3 6 3 3 3 3 3 11 6 6 10 15 3 3 10 3.04 3.15 3.37 3.23 3.84 3.74 3.55 3.47 3.49 3.06 3.23 3.36 3.13 3.24 3.61 3.77 3.52 3.62 3.68 3.57 3.55 3.61 3.75 3.72 3.76 3.71 3.68 3.66 3.70 3.45 3.67 3.54 3.52 3.66 3.52 3.61 3.67 3.54 3.65 3.55 3.49 3.51 3.56 3.57 3.48 3.70 3.51 3.59 3.58 3.55 3.52 3.65 3.62 3.57 3.63 3.64 3.55 3.65 3.56 3.57 3.20 3.41 3.24 3.24 4.55 4.76 3.08 4.11 3.33 3.37 3.44 3.92 3.24 3.55 3.54 4.49 3.58 4.47 4.57 4.36 4.63 4.20 3.26 4.64 3.85 3.15 3.33 4.24 4.41 3.59 3.25 3.89 3.11 3.68 3.77 3.66 3.27 3.81 3.98 3.77 3.70 3.82 3.48 3.89 3.91 3.36 3.37 3.33 3.53 3.51 3.66 3.51 3.54 3.36 3.70 3.42 3.53 3.24 3.49 3.44 3.78 4.01 3.50 3.94 4.11 4.05 3.74 4.04 4.16 3.96 3.93 4.05 3.94 4.10 4.10 3.75 3.76 3.63 3.79 3.80 3.89 3.89 4.03 3.72 4.00 3.99 3.96 3.63 3.81 3.86 3.92 3.93 3.82 3.99 4.04 4.35 3.82 3.89 3.76 3.83 3.90 4.00 3.79 3.80 3.67 3.65 3.74 3.51 3.76 3.73 3.86 3.79 3.82 3.56 3.82 3.82 3.83 3.72 3.85 3.72 2.11 2.08 2.08 2.00 2.49 2.58 2.02 2.38 2.06 2.05 2.10 2.19 2.06 2.10 2.19 2.71 2.24 2.75 2.69 2.60 2.70 2.50 2.09 2.79 2.33 2.08 2.17 2.51 2.59 2.27 1 4 1 2 3 4 1 4 1 3 3 4 1 3 2 4 4 4 4 4 4 4 3 4 3 3 4 4 4 4 33 25 34 51 6 27 32 22 38 28 75 19 29 47 43 21 46 31 30 25 31 25 60 34 49 64 93 24 28 57 149 275 349 920 12 83 92 128 616 176 673 281 204 687 912 124 244 360 232 188 258 170 440 347 386 587 914 154 88 307 37.25 68.75 87.25 76.66667 12 20.75 30.66667 64 56 35.2 44.86667 70.25 40.8 85.875 182.4 41.33333 40.66667 120 77.33333 62.66667 86 56.66667 40 57.83333 64.33333 58.7 60.93333 51.33333 29.33333 30.7 Latitude and longitude do not influence species richness. Is species richness correlated with longitude and latitude? Does the distance between islands influence species richness? Are geographically near islands also similar in species richness irrespective of island area? R(S-Long) = 0.22 n.s. R(S-Lat) = 0.28 n.s.) That there is no significant correlation does not mean that latitude and longitude do not have an influence on the regression model with environmental variables. Spatial autocorrelation S2 S1 S3 S4 S5 S6 In spatial autocorrelation the distance between study sites influence the response (dependent) variable. Spatialy adjacent sites are then expected to be more similar with respect to the response variable. Moran’s I as a measure of spatial autocorrelation Moran’s I is similar to a correlation coefficient all applied to pairwise cells of a spatial matrix. It differs by weighting the covariance to account for spatial non-independence of cells with respect to distance. S2 S1 N S3 I S4 S1 2 4 2 5 3 3 i 1 j1 N N w i 1 j1 S5 w z z N N N ij z i 1 ij i j w ij 2 i 1 (1 d ij ) 2 S6 S2 3 6 4 6 6 5 Distance 0.3 0.4 0.7 0.2 0.9 0.6 All combinations of sites If cell values were randomly distributed (not spatially autocorrelated) the expected I is E 0 (I) 1 N 1 Statistical significance is calculated from a Monte Carlo simulation Individuals/trap is slightly spatially autocorrelated Latitude and longitude slightly influence species richenss. Even this weak effect might influence the outcome of a regression analysis. Log transformed variables Akaike information criterion. The lower AIC the more appropriate is the model Errors: Too many variables!! Solution: prior factor analysis to reduce the number of dependent variables OLS result Spatial autoregression result Stepwsie variable reduction Information criteria 3300 3000 2 y = 16.113x - 185.67x + 533.84 R² = 0.8953 2500 2800 2300 2000 1500 Y Y y = -0.375x4 + 14.462x3 - 164.12x2 + 609.02x - 356.84 R² = 0.9607 1800 1300 1000 800 y = 45.502e0.1932x R² = 0.5286 500 0 0 5 10 X 15 y = 90.901x R² = 0.5747 300 -200 20 0 5 What function fits best? The more free parameters a model has the higher will be R2. The more parsimonious a model is the lesser is the bias towards type I errors. 10 15 20 X Explained variance Bias We have to find a compromis between goodness of fit and bias! many few Model parameters The optimal number of model parameters The Akaike criterion of model choice AIC 2k 2 ln( L) k: number of model parameters L: maximum likelihood estimate of the model The preferred model is the one with the lowest AIC. If the parameter errors are AIC normal and independent we get 2k n ln( RSS / n) 2k n ln( RSS ) n ln( n) n: number data points RSS: residual sums of squares n RSS i If we fit using 2: AIC 2 2k 2 i 1 2 If we fit using R2: AICR 2 1 R2 2k ln n At small sample size we should use the following correction AIC C AIC 2k (k 1) nk 2 3300 3000 2 y = 16.113x - 185.67x + 533.84 R² = 0.8953 2500 2800 2300 2000 1500 Y Y y = -0.375x4 + 14.462x3 - 164.12x2 + 609.02x - 356.84 R² = 0.9607 1800 1300 1000 800 y = 45.502e0.1932x 500 R² = 0.5286 0 0 AICC R 2 5 10 X 15 1 R 2 2k (k 1) 2k ln n nk 2 We get the surprising result that the seemingly worst fitting model appears to be the preferred one. A single outlier makes the difference. The single high residual makes the exponential fitting worse -200 20 y = 90.901x R² = 0.5747 300 0 5 10 15 20 X 1 0.8953 6(3 1) AICC R 2 6 ln 2.51 19 19 3 2 1 0.5286 4(2 1) AICC R 2 4 ln 1.10 19 19 2 2 1 0.9607 10(5 1) AICC R 2 10 ln 8.82 19 19 5 2 1 0.5747 2(1 1) AICC R 2 2 ln 1.56 19 19 1 2 Significant difference in model fit AIC AIC1 AIC2 Approximately AIC is statisticaly significant in favor of the model with thesmaller AIC at the 5% error benchmark if |AIC| > 2. 1 0.8953 6(3 1) AICC R 2 6 ln 2.51 19 19 3 2 1 0.5286 4(2 1) AICC R 2 4 ln 1.10 19 19 2 2 1 0.9607 10(5 1) AICC R 2 10 ln 8.82 19 19 5 2 1 0.5747 2(1 1) AICC R 2 2 ln 1.56 19 19 1 2 AIC 1.56 1.10 2.66 The last model is significantly (5% level) the best. Stepwise variable elimination Standardized coefficients (b-values) are equivalents of correlation coefficients. They should have values above 1. Such values point to too high correlation between the predictor variables (collinearity). Collnearity disturbs any regression model and has to be eliminated prior to analysis. Highly correlated variables essentially contain the same information. Correlations of less than 0.7 can be tolerated. Hence check first the matrix of correlation coefficients. Eliminate variables that do not add information. The final model Simple test wise probability levels. We yet have to correct for multiple testing. The best model is not always the one with the lowest AIC or the highest R2. Bonferroni correction p ( I ) p ( I ) 1 p(I n ) 1 n p( I n ) 1 1 n p( I n ) 1 (1 n ) n p( I n ) n To get an experiment wise error rate of 0.05 our test wise error rates have be less than 0.05/n Species richness is positively correlated with island area and negatively with soil humidity. Island ensembles differ in species richness and abundance. A simple ANOVA does not detect any difference Species richness depends on environmental factors that may differ between island ensembles. Analysis of covariance (ANCOVA) Analysis of covariance (ANCOVA) ANCOVA is the combination of multiple regression and analysis of variance. First we perform a regression anlyis and use the residuals of the full model as entries in the ANOVA. ANCOVA is the ANOVA on regression residuals. 100.0 y = 0.9377x + 2.6159 R² = 0.843 90.0 The metrically scaled variables serve as covariates. Sites with very high positive residuals are particularly species rich even after controlling for environmental factors. These are ecological hot spots. Regression analysis serves to identify such hot spots We use the regression residuals for further analysis Observed value 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 0 20 40 60 80 Predicted value 100 120 ANCOVA Islands IslandEns emble MNB GórnaE MNB Koń MNB Kopanka MNB KrólewskiOstrów MNB Maleńka MałaWierzba MNB MNB KopankaN MNB Ośrodek MNB Piaseczna MNB Ruciane MNB Mikołajki MNB Śluza MNB GórnaW MNB Wierzba Wygryńska MNB Wigry Bryzgiel Wigry Bryzgiel BrzozowaL Wigry BrzozowaP Wigry Wigry CimochowskiGrądzikC Wigry CimochowskiGrądzikN Wigry CimochowskiGrądzikS Wigry Kamień Wigry Krowa MysiaWigry Wigry Wigry Ordów Wigry Ostrów Wigry Rośków Wigry Walędziak WysokiWęgiełWigry Area 0.70 0.50 0.69 6.15 0.00 0.40 0.18 0.09 0.63 10.00 10.00 0.48 0.44 0.78 0.67 0.20 10.00 3.81 2.32 0.15 0.14 0.76 3.13 4.49 1.55 8.69 38.82 0.56 0.76 10.0 Organ SoilD Temp SoilH Individuals icMatt Distur Speci SoilF SoilA Individ disper Traps Light eratur umidit /trap eCont bance es ertility cidity sion y e ent 4.0 4.0 4.0 12.0 1.0 4.0 3.0 2.0 11.0 5.0 15.0 4.0 5.0 8.0 5.0 3.0 6.0 3.0 3.0 3.0 3.0 3.0 11.0 6.0 6.0 10.0 15.0 3.0 3.0 10 3.04 3.15 3.37 3.23 3.84 3.74 3.55 3.47 3.49 3.06 3.23 3.36 3.13 3.24 3.61 3.77 3.52 3.62 3.68 3.57 3.55 3.61 3.75 3.72 3.76 3.71 3.68 3.66 3.70 3.45 3.67 3.54 3.52 3.66 3.52 3.61 3.67 3.54 3.65 3.55 3.49 3.51 3.56 3.57 3.48 3.70 3.51 3.59 3.58 3.55 3.52 3.65 3.62 3.57 3.63 3.64 3.55 3.65 3.56 3.57 3.20 3.41 3.24 3.24 4.55 4.76 3.08 4.11 3.33 3.37 3.44 3.92 3.24 3.55 3.54 4.49 3.58 4.47 4.57 4.36 4.63 4.20 3.26 4.64 3.85 3.15 3.33 4.24 4.41 3.59 3.25 3.89 3.11 3.68 3.77 3.66 3.27 3.81 3.98 3.77 3.70 3.82 3.48 3.89 3.91 3.36 3.37 3.33 3.53 3.51 3.66 3.51 3.54 3.36 3.70 3.42 3.53 3.24 3.49 3.44 3.78 4.01 3.50 3.94 4.11 4.05 3.74 4.04 4.16 3.96 3.93 4.05 3.94 4.10 4.10 3.75 3.76 3.63 3.79 3.80 3.89 3.89 4.03 3.72 4.00 3.99 3.96 3.63 3.81 3.86 3.92 3.93 3.82 3.99 4.04 4.35 3.82 3.89 3.76 3.83 3.90 4.00 3.79 3.80 3.67 3.65 3.74 3.51 3.76 3.73 3.86 3.79 3.82 3.56 3.82 3.82 3.83 3.72 3.85 3.72 2.11 2.08 2.08 2.00 2.49 2.58 2.02 2.38 2.06 2.05 2.10 2.19 2.06 2.10 2.19 2.71 2.24 2.75 2.69 2.60 2.70 2.50 2.09 2.79 2.33 2.08 2.17 2.51 2.59 2.27 1.0 4.0 1.0 2.0 3.0 4.0 1.0 4.0 1.0 3.0 3.0 4.0 1.0 3.0 2.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 3.0 4.0 3.0 3.0 4.0 4.0 4.0 4.0 33.0 25.0 34.0 51.0 6.0 27.0 32.0 22.0 38.0 28.0 75.0 19.0 29.0 47.0 43.0 21.0 46.0 31.0 30.0 25.0 31.0 25.0 60.0 34.0 49.0 64.0 93.0 24.0 28.0 57.0 149.0 275.0 349.0 920.0 12.0 83.0 92.0 128.0 616.0 176.0 673.0 281.0 204.0 687.0 912.0 124.0 244.0 360.0 232.0 188.0 258.0 170.0 440.0 347.0 386.0 587.0 914.0 154.0 88.0 307.0 37.3 68.8 87.3 76.7 12.0 20.8 30.7 64.0 56.0 35.2 44.9 70.3 40.8 85.9 182.4 41.3 40.7 120.0 77.3 62.7 86.0 56.7 40.0 57.8 64.3 58.7 60.9 51.3 29.3 30.7 Species richness does not differ between island ensembles. Model Residuals 3.569704 3.468026 3.500137 3.963326 1.854909 3.173658 3.391454 2.977121 3.719825 3.823174 4.003134 3.079715 3.388073 3.29098 3.754503 3.076486 3.90665 3.623234 3.568952 3.060985 3.127977 3.322735 4.025703 3.593559 3.761767 4.248705 4.584715 3.041533 3.390937 3.955794 -0.073 -0.249 0.026 -0.032 -0.063 0.122 0.074 0.114 -0.082 -0.491 0.314 -0.135 -0.021 0.559 0.007 -0.032 -0.078 -0.189 -0.168 0.158 0.306 -0.104 0.069 -0.067 0.130 -0.090 -0.052 0.137 -0.059 0.087 • The degree of disturbance (human impact) influences species richenss. 120.0 Observed value 100.0 y = 0.9243x + 3.3687 R² = 0.8364 80.0 60.0 40.0 20.0 0.0 0 20 40 60 80 100 120 Predicted value Species richness of spiders on lake islands appears to be independent of the degree of disturbance How does abundance depend on environmental fatcors? The ful model and stepwise variable elimination All coefficients are highly significant! All standardized coefficients are above 1. This points to too high collinearity We furthr eliminate uninformative variables. Abundance does not significally depend on environmental variables How does abundance depend on the degree of disturbance? Abundance of spiders on lake islands appears to be independent of the degree of disturbance