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Statistical Analysis Chi Square (X2)
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Chi Square - a test of statistical significance
Chi Square measures whether any deviations from the predicted norm that occurs in experimental results exceeds the deviation that might occur by chance. If it is found to be consistent with chance then it can be said to be due to genetics. X2 = ∑( d2 / e ) Chi square deviation expected Sum squared
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Flip a coin 100 times –. you expect 50. heads and 50 tails
Flip a coin 100 times – you expect heads and 50 tails X2 = ∑( d2 / e ) Heads Tails Observed values 45 55 Expected (e) 50 Deviation (d) 5 Deviation squared (d2) 25 d2 / e 25/50 = 0.5 ∑( d2 / e ) = 1
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Flip a coin 500 times –. you expect 250. heads and 250 tails
Flip a coin 500 times – you expect heads and 250 tails X2 = ∑( d2 / e ) Heads Tails Observed values 245 255 Expected (e) 250 Deviation (d) 5 Deviation squared (d2) 25 d2 / e 25/250 = 0.1 ∑( d2 / e ) = 0.2
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Sample Size Notice that the sample size is very important. The larger your sample size, the closer your data should be to the expected data if the trait you are examining is due to chance (as genetics should be).
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Phenotypes The number of possible phenotypes is also very important. In the case of coin flipping, there are 2 “phenotypes” In statistics we refer to this as having one degree of freedom. The degrees of freedom is on less than the total number of possible phenotypes. 4 phenotypes means 3 degrees of freedom 6 phenotypes means 5 degrees of freedom
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How we use this information
How we use this information - refer to the table This table gives the probability that an amount of deviation would occur simply by chance. df 0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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From our first coin toss example, X2 = 1 degrees of freedom (df) = 1 Therefore there is about a 30% probability that this was due to chance. This is not considered statistically significant. df 0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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From our second coin toss example, X2 = 0
From our second coin toss example, X2 = 0.2 degrees of freedom (df) = 1 Therefore there is less than 70% probability that this was due to chance. This is not considered statistically significant. But it is better! df 0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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Coins are not Genetics In most scientific research we do not want the results to be due to chance so for data to be considered statistically significant, the X2 result must be 5% or less. In genetics we want to find consistent data that shows high probability due to chance so the higher the probability due to chance, the better. To be considered statistically significant, the X2 result must be 95% or more.
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values Expected (e) Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) 56 19 Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) 56 19 Deviation (d) 5 3 2 Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) 56 19 Deviation (d) 5 3 2 Deviation squared (d2) 25 9 4 d2 / e ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) 56 19 Deviation (d) 5 3 2 Deviation squared (d2) 25 9 4 d2 / e 0.45 0.16 0.21 ∑( d2 / e )
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Looking at our corn cobs One group had the following data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 169 61 53 17 300 RP Rp rP rp Observed values 169 61 53 17 Expected (e) 56 19 Deviation (d) 5 3 2 Deviation squared (d2) 25 9 4 d2 / e 0.45 0.16 0.21 ∑( d2 / e ) = 0.82
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X2 = 0.82 degrees of freedom (df) = 3
0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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X2 = degrees of freedom (df) = 3 Therefore there is about 85% probability that this was due to chance. This is not considered statistically significant. df 0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values Expected (e) Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) 2226 742 247 Deviation (d) Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) 2226 742 247 Deviation (d) 152 82 81 11 Deviation squared (d2) d2 / e ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) 2226 742 247 Deviation (d) 152 82 81 11 Deviation squared (d2) 23104 6724 6561 121 d2 / e ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) 2226 742 247 Deviation (d) 152 82 81 11 Deviation squared (d2) 23104 6724 6561 121 d2 / e 10.38 9.06 8.8 0.49 ∑( d2 / e )
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Looking at the corn cob class data
Round, Purple Round, Yellow Wrinkled, Purple Wrinkled, Yellow Total 2378 660 661 258 3957 RP Rp rP rp Observed values 2378 660 661 258 Expected (e) 2226 742 247 Deviation (d) 152 82 81 11 Deviation squared (d2) 23104 6724 6561 121 d2 / e 10.38 9.06 8.8 0.49 ∑( d2 / e ) = 28.73
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X2 = 28.73 degrees of freedom (df) = 3
0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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X2 = degrees of freedom (df) = 3 Therefore there is less than 1% probability that this was due to chance. This is really bad data!!! df 0.99 0.97 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01 1 --- 0.001 0.004 0.016 0.060 0.15 0.46 1.07 1.64 2.706 3.841 5.024 6.635 2 0.051 0.103 0.211 0.45 0.71 1.39 2.41 3.22 4.605 5.991 7.378 9.210 3 0.11 0.216 0.352 0.584 1.01 1.42 2.37 3.66 4.64 6.251 7.815 9.348 11.34 4 0.29 0.484 0.711 1.064 1.65 2.20 3.36 4.88 5.99 7.779 9.488 11.14 13.27 5 0.55 0.831 1.145 1.610 2.34 3.00 4.35 6.06 7.29 9.236 11.07 12.83 15.08
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Chi-Square Test Problems question b)
Green Yellow Blue Observed values 19 18 37 Expected (e) 18.5 Deviation (d) 0.5 Deviation squared (d2) 0.25 d2 / e 0.0135 ∑( d2 / e ) = 0.027 This indicates just under 99% probability due to chance. Therefore we can accept this as statistically significant.
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Chi-Square Test Problems question c)
RP Rp rP rp Observed values 216 65 79 21 Expected (e) 214 71 24 Deviation (d) 2 6 8 3 Deviation squared (d2) 4 36 64 9 d2 / e 0.019 0.51 0.90 0.375 ∑( d2 / e ) = 1.804 This indicates about 65% probability due to chance. Therefore we can not accept this as statistically significant.
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Chi-Square Test Problems question d)
Ty tY TY ty Observed values 193 184 556 61 Expected (e) 186 559 62 Deviation (d) 7 2 3 1 Deviation squared (d2) 49 4 9 d2 / e 0.26 0.02 ∑( d2 / e ) = 0.32 This indicates about 96% probability due to chance. Therefore we can accept this as statistically significant.
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