Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka pilina manawa huahana
ʻO ka Product-Moment Correlation, i ʻike ʻia hoʻi ʻo Pearson Correlation, he ʻano hana helu i hoʻohana ʻia e ana i ka ikaika a me ke kuhikuhi o kahi pilina linear ma waena o ʻelua mau loli. He mea pono kēia ʻano hana ma nā ʻano like ʻole, mai ka noiʻi hoʻonaʻauao a me ka loiloi ʻoihana a hiki i ka loiloi ʻana o nā hoʻokolohua ma nā ʻepekema kūlohelohe. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā no ka helu ʻana i ka Product-Moment Correlation.
Pendahuluan
Ma mua o ko mākou komo ʻana i nā nīnau hoʻohālike, he manaʻo maikaʻi e hoʻomaopopo i ke kumumanaʻo kumu o ka Product Moment Correlation. ʻO ke ʻano maʻamau i hoʻohana ʻia e helu ai i ka Pearson correlation coefficient (\(r\)) penei:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
Ma hea:
– ʻO \( n \) ka helu o nā hui ʻikepili.
– ʻO \( \sum{XY} \) ka huina o nā huahana o \( X \) a me \( Y \).
– ʻO \( \sum{X} \) ka huina o nā loli \( X \).
– ʻO \( \sum{Y} \) ka huina o nā loli \( Y \).
– ʻO \( \sum{X^2} \) ka huina o nā huinahā o ka loli \( X \).
– ʻO \( \sum{Y^2} \) ka huina o nā huinahā o ka loli \( Y \).
Aia mau ka helu hoʻohālikelike o Pearson (\( r \)) ma waena o -1 a me 1. Hōʻike ka pilina maikaʻi e neʻe like nā loli ʻelua, ʻoiai ʻo ka pilina maikaʻi ʻole e hōʻike ana i ka piʻi ʻana o kekahi loli, e emi ana kekahi. Inā \( r = 0 \), a laila ʻaʻohe pilina linear ma waena o nā loli ʻelua.
Laʻana Nīnau 1
'Ikepili
Eia ka ʻikepili helu hoʻokolohua makemakika a me ke kino no 5 mau haumāna:
| Haumāna | Makemakika (X) | ʻEpekema Kino (Y) |
|——-|———————-|—————-|
| 1 | 85 | 90 |
| 2 | 78 | 85 |
| 3 | 85 | 80 |
| 4 | 70 | 70 |
| 5 | 80 | 88 |
Nā ʻanuʻu hoʻonā
1. Ke helu ʻana i nā ʻāpana koʻikoʻi:
– \( \sum{X} \) = 85 + 78 + 85 + 70 + 80 = 398
– \( \sum{Y} \) = 90 + 85 + 80 + 70 + 88 = 413
– \( \sum{XY} \) = (85\ 90) + (78\ 85) + (85\ 80) + (70\ 70) + (80\ 88) = 7650 + 6630 + 6800 + 4900 + 7040 = 33020
– \( \sum{X^2} \) = (85^2) + (78^2) + (85^2) + (70^2) + (80^2) = 7225 + 6084 + 7225 + 4900 + 6400 = 31834
– \( \sum{Y^2} \) = (90^2) + (85^2) + (80^2) + (70^2) + (88^2) = 8100 + 7225 + 6400 + 4900 + 7744 = 34369
2. E komo i loko o ke ʻano hana:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{5(33020) – (398)(413)}{\sqrt{[5(31834) – (398)^2][5(34369) – (413)^2]}} \]
3. Ke helu ʻana i nā hopena:
– Heluhelu: \( 5(33020) – (398)(413) = 165100 – 164474 = 626 \)
– Mea hoʻokaʻawale:
– \( n\sum{X^2} – (\sum{X})^2 = 5(31834) – (398)^2 = 159170 – 158404 = 766 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 5(34369) – (413)^2 = 171845 – 170569 = 1276 \)
– \( \sqrt{766 \times 1276} \approx \sqrt{976856} \approx 989.36 \)
\[ r = \frac{626}{989.36} \approx 0.633 \]
No laila, ʻo ke koina pilina Pearson ma waena o nā helu hoʻāʻo makemakika a me ke kino he 0.633, e hōʻike ana he pilina maikaʻi kūpono ma waena o nā loli ʻelua.
Laʻana Nīnau 2
'Ikepili
Eia ka ʻikepili e pili ana i ke kumukūʻai kūʻai aku a me nā lilo hoʻolaha mai 6 mau mahina ma kahi ʻoihana:
| Mahina | Hoʻolaha (X) | Kūʻai aku (Y) |
|——-|—————–|——————|
| 1 | 2000 | 2500 |
| 2 | 1800 | 2100 |
| 3 | 2200 | 2700 |
| 4 | 2400 | 2900 |
| 5 | 2300 | 3000 |
| 6 | 2500 | 3200 |
Nā ʻanuʻu hoʻonā
1. Ke helu ʻana i nā ʻāpana koʻikoʻi:
– \( \sum{X} \) = 2000 + 1800 + 2200 + 2400 + 2300 + 2500 = 13200
– \( \sum{Y} \) = 2500 + 2100 + 2700 + 2900 + 3000 + 3200 = 16400
– \( \sum{XY} \) = (2000\ 2500) + (1800\ 2100) + (2200\ 2700) + (2400\ 2900) + (2300\ 3000) + (2500\ 3200) = 5000000 + 3780000 + 5940000 + 6960000 + 6900000 + 8000000 = 36580000
– \( \sum{X^2} \) = (2000^2) + (1800^2) + (2200^2) + (2400^2) + (2300^2) + (2500^2) = 4000000 + 3240000 + 4840000 + 5760000 + 5290000 + 6250000 = 29380000
– \( \sum{Y^2} \) = (2500^2) + (2100^2) + (2700^2) + (2900^2) + (3000^2) + (3200^2) = 6250000 + 4410000 + 7290000 + 8410000 + 9000000 + 10240000 = 45590000
2. E komo i loko o ke ʻano hana:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{6(36580000) – (13200)(16400)}{\sqrt{[6(29380000) – (13200)^2][6(45590000) – (16400)^2]}} \]
3. Ke helu ʻana i nā hopena:
– Heluhelu: \( 6(36580000) – (13200)(16400) = 219480000 – 216480000 = 3000000 \)
– Mea hoʻokaʻawale:
– \( n\sum{X^2} – (\sum{X})^2 = 6(29380000) – (13200)^2 = 176280000 – 174240000 = 2040000 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 6(45590000) – (16400)^2 = 273540000 – 268960000 = 4580000 \)
– \( \sqrt{2040000 \times 4580000} \approx \sqrt{9343200000000} \approx 3056246.20 \)
\[ r = \frac{3000000}{3056246.20} \approx 0.981 \]
No laila, ʻo ke koina pilina Pearson ma waena o ka hoʻolilo hoʻolaha a me ke kumukūʻai kūʻai aku he 0.981, e hōʻike ana he pilina maikaʻi loa ma waena o nā loli ʻelua.
Ka hopena
He mea hana pono loa ka Pearson correlation coefficient (\(r\)) no ka hoʻomaopopo ʻana i ka pilina linear ma waena o ʻelua mau loli. Ma nā laʻana i hāʻawi ʻia, ʻike mākou pehea e helu ai i ka waiwai \(r\) a wehewehe iā ia. ʻO ka pilina kiʻekiʻe (kokoke i ka 1 a i ʻole -1) e hōʻike ana i ka pilina ikaika, ʻoiai ʻo ka pilina haʻahaʻa (kokoke i ka 0) e hōʻike ana i ka pilina nāwaliwali. He mea nui e hoʻomaopopo ʻaʻole hōʻike ka pilina i ke kumu; hōʻike wale ia aia kahi pilina ma waena o nā loli ʻelua.
