I'm testing for normal distribution, but different test are giving me different outputs and I don't know which one I should believe/use.
Another problem is, that I don't know if I should use the fixed effects model or the stadard model, without anything to check for nomal distribution. The fixed effects model is the one I have to use later, so I thought it would be more important to know what's going on with this model or am I wrong?
The CODE is the following:
FORMULAR.PLM.BASELINE <- StockPrice.WINS ~ EPS.WINS BookValuePS.WINS ESGscore.WINS ggroup
MOD.FE <- plm(FORMULAR.PLM.BASELINE, data = PD.Datensatz_final, model = "within", effect = "twoways")
summary(MOD.FE)
VEC.Residuals.FE <- residuals(MOD.FE)
# The strange thing is that there are different outcomes influenced by the test I use. How do I decide which one to use?
# Outcome behind the test
# Shapiro-Wilk-Test #
shapiro.test(VEC.Residuals) # NO normal distribution
# Chi-Square-Test #
pearson.test(VEC.Residuals, adjust = FALSE) # normal distribution
# Kolmogoroff-Smirnov-Lillefors-Test #
lillie.test(VEC.Residuals) # normal distribution
# Shapiro-Francia-Test #
sf.test(VEC.Residuals) # NO normal distribution
# Anderson-Darling-Test #
ad.test(VEC.Residuals) # normal distribution
DATA:
"AIR PRODUCTS & CHEMICALS INC", "AIR PRODUCTS & CHEMICALS INC",
"AIR PRODUCTS & CHEMICALS INC", "AIR PRODUCTS & CHEMICALS INC",
"AIR PRODUCTS & CHEMICALS INC", "HESS CORP", "HESS CORP", "HESS CORP",
"HESS CORP", "HESS CORP", "HESS CORP", "HESS CORP", "APACHE CORP",
"APACHE CORP", "APACHE CORP", "APACHE CORP", "APACHE CORP", "APACHE CORP",
"APACHE CORP", "AVERY DENNISON CORP", "AVERY DENNISON CORP",
"AVERY DENNISON CORP", "AVERY DENNISON CORP", "AVERY DENNISON CORP",
"AVERY DENNISON CORP", "AVERY DENNISON CORP", "BALL CORP", "BALL CORP",
"BALL CORP", "BALL CORP", "BALL CORP", "BALL CORP", "BALL CORP",
"CHEVRON CORP", "CHEVRON CORP", "CHEVRON CORP", "CHEVRON CORP",
"CHEVRON CORP", "CHEVRON CORP", "CHEVRON CORP", "ECOLAB INC",
"ECOLAB INC", "ECOLAB INC", "ECOLAB INC", "ECOLAB INC", "ECOLAB INC",
"ECOLAB INC", "EXXON MOBIL CORP", "EXXON MOBIL CORP", "EXXON MOBIL CORP",
"EXXON MOBIL CORP", "EXXON MOBIL CORP", "EXXON MOBIL CORP", "EXXON MOBIL CORP",
"FMC CORP", "FMC CORP", "FMC CORP", "FMC CORP", "FMC CORP", "FMC CORP",
"FMC CORP", "HALLIBURTON CO", "HALLIBURTON CO", "HALLIBURTON CO",
"HALLIBURTON CO", "HALLIBURTON CO", "HALLIBURTON CO", "HALLIBURTON CO",
"HELMERICH & PAYNE", "HELMERICH & PAYNE", "HELMERICH & PAYNE",
"HELMERICH & PAYNE", "HELMERICH & PAYNE", "HELMERICH & PAYNE",
"HELMERICH & PAYNE", "HOLLYFRONTIER CORP", "HOLLYFRONTIER CORP",
"HOLLYFRONTIER CORP", "HOLLYFRONTIER CORP", "HOLLYFRONTIER CORP",
"HOLLYFRONTIER CORP", "HOLLYFRONTIER CORP", "INTL FLAVORS & FRAGRANCES",
"INTL FLAVORS & FRAGRANCES", "INTL FLAVORS & FRAGRANCES", "INTL FLAVORS & FRAGRANCES",
"INTL FLAVORS & FRAGRANCES", "INTL FLAVORS & FRAGRANCES", "INTL FLAVORS & FRAGRANCES",
"INTL PAPER CO", "INTL PAPER CO", "INTL PAPER CO", "INTL PAPER CO",
"INTL PAPER CO", "INTL PAPER CO", "INTL PAPER CO", "MARATHON OIL CORP",
"MARATHON OIL CORP", "MARATHON OIL CORP", "MARATHON OIL CORP",
"MARATHON OIL CORP", "MARATHON OIL CORP", "MARATHON OIL CORP",
"NEWMONT CORP", "NEWMONT CORP", "NEWMONT CORP", "NEWMONT CORP",
"NEWMONT CORP", "NEWMONT CORP", "NEWMONT CORP", "NUCOR CORP",
"NUCOR CORP", "NUCOR CORP", "NUCOR CORP", "NUCOR CORP", "NUCOR CORP",
"NUCOR CORP", "OCCIDENTAL PETROLEUM CORP", "OCCIDENTAL PETROLEUM CORP",
"OCCIDENTAL PETROLEUM CORP", "OCCIDENTAL PETROLEUM CORP", "OCCIDENTAL PETROLEUM CORP",
"OCCIDENTAL PETROLEUM CORP", "OCCIDENTAL PETROLEUM CORP", "ONEOK INC",
"ONEOK INC", "ONEOK INC", "ONEOK INC", "ONEOK INC", "ONEOK INC",
"ONEOK INC", "PPG INDUSTRIES INC", "PPG INDUSTRIES INC", "PPG INDUSTRIES INC",
"PPG INDUSTRIES INC", "PPG INDUSTRIES INC", "PPG INDUSTRIES INC",
"PPG INDUSTRIES INC", "CONOCOPHILLIPS", "CONOCOPHILLIPS", "CONOCOPHILLIPS",
"CONOCOPHILLIPS", "CONOCOPHILLIPS", "CONOCOPHILLIPS", "CONOCOPHILLIPS",
"SCHLUMBERGER LTD", "SCHLUMBERGER LTD", "SCHLUMBERGER LTD", "SCHLUMBERGER LTD",
"SCHLUMBERGER LTD", "SCHLUMBERGER LTD", "SCHLUMBERGER LTD", "SHERWIN-WILLIAMS CO",
"SHERWIN-WILLIAMS CO", "SHERWIN-WILLIAMS CO", "SHERWIN-WILLIAMS CO",
"SHERWIN-WILLIAMS CO", "SHERWIN-WILLIAMS CO", "SHERWIN-WILLIAMS CO",
"VULCAN MATERIALS CO", "VULCAN MATERIALS CO", "VULCAN MATERIALS CO",
"VULCAN MATERIALS CO", "VULCAN MATERIALS CO", "VULCAN MATERIALS CO",
"VULCAN MATERIALS CO", "WILLIAMS COS INC", "WILLIAMS COS INC",
"WILLIAMS COS INC", "WILLIAMS COS INC", "WILLIAMS COS INC", "WILLIAMS COS INC",
"WILLIAMS COS INC", "PIONEER NATURAL RESOURCES CO", "PIONEER NATURAL RESOURCES CO",
"PIONEER NATURAL RESOURCES CO", "PIONEER NATURAL RESOURCES CO",
"PIONEER NATURAL RESOURCES CO", "PIONEER NATURAL RESOURCES CO",
"PIONEER NATURAL RESOURCES CO", "FREEPORT-MCMORAN INC", "FREEPORT-MCMORAN INC",
"FREEPORT-MCMORAN INC", "FREEPORT-MCMORAN INC", "FREEPORT-MCMORAN INC",
"FREEPORT-MCMORAN INC", "FREEPORT-MCMORAN INC", "DEVON ENERGY CORP",
"DEVON ENERGY CORP", "DEVON ENERGY CORP", "DEVON ENERGY CORP",
"DEVON ENERGY CORP", "DEVON ENERGY CORP", "DEVON ENERGY CORP",
"VALERO ENERGY CORP", "VALERO ENERGY CORP", "VALERO ENERGY CORP",
"VALERO ENERGY CORP", "VALERO ENERGY CORP", "VALERO ENERGY CORP",
"VALERO ENERGY CORP", "EOG RESOURCES INC", "EOG RESOURCES INC",
"EOG RESOURCES INC", "EOG RESOURCES INC", "EOG RESOURCES INC",
"EOG RESOURCES INC", "EOG RESOURCES INC", "CABOT OIL & GAS CORP",
"CABOT OIL & GAS CORP", "CABOT OIL & GAS CORP", "CABOT OIL & GAS CORP",
"CABOT OIL & GAS CORP", "CABOT OIL & GAS CORP", "CABOT OIL & GAS CORP",
"LINDE PLC", "LINDE PLC", "LINDE PLC", "LINDE PLC", "LINDE PLC",
"LINDE PLC", "LINDE PLC", "EASTMAN CHEMICAL CO", "EASTMAN CHEMICAL CO",
"EASTMAN CHEMICAL CO", "EASTMAN CHEMICAL CO", "EASTMAN CHEMICAL CO",
"EASTMAN CHEMICAL CO", "EASTMAN CHEMICAL CO", "MARTIN MARIETTA MATERIALS",
"MARTIN MARIETTA MATERIALS", "MARTIN MARIETTA MATERIALS", "MARTIN MARIETTA MATERIALS",
"MARTIN MARIETTA MATERIALS", "MARTIN MARIETTA MATERIALS", "MARTIN MARIETTA MATERIALS",
"ALBEMARLE CORP", "ALBEMARLE CORP", "ALBEMARLE CORP", "ALBEMARLE CORP",
"ALBEMARLE CORP", "ALBEMARLE CORP", "ALBEMARLE CORP", "NATIONAL OILWELL VARCO INC",
"NATIONAL OILWELL VARCO INC", "NATIONAL OILWELL VARCO INC", "NATIONAL OILWELL VARCO INC",
"NATIONAL OILWELL VARCO INC", "NATIONAL OILWELL VARCO INC", "NATIONAL OILWELL VARCO INC",
"PACKAGING CORP OF AMERICA", "PACKAGING CORP OF AMERICA", "PACKAGING CORP OF AMERICA",
"PACKAGING CORP OF AMERICA", "PACKAGING CORP OF AMERICA", "PACKAGING CORP OF AMERICA",
"PACKAGING CORP OF AMERICA", "MOSAIC CO", "MOSAIC CO", "MOSAIC CO",
"MOSAIC CO", "MOSAIC CO", "MOSAIC CO", "MOSAIC CO", "CELANESE CORP",
"CELANESE CORP", "CELANESE CORP", "CELANESE CORP", "CELANESE CORP",
"CELANESE CORP", "CELANESE CORP", "CF INDUSTRIES HOLDINGS INC",
"CF INDUSTRIES HOLDINGS INC", "CF INDUSTRIES HOLDINGS INC", "CF INDUSTRIES HOLDINGS INC",
"CF INDUSTRIES HOLDINGS INC", "CF INDUSTRIES HOLDINGS INC", "CF INDUSTRIES HOLDINGS INC",
"LYONDELLBASELL INDUSTRIES NV", "LYONDELLBASELL INDUSTRIES NV",
"LYONDELLBASELL INDUSTRIES NV", "LYONDELLBASELL INDUSTRIES NV",
"LYONDELLBASELL INDUSTRIES NV", "LYONDELLBASELL INDUSTRIES NV",
"LYONDELLBASELL INDUSTRIES NV"), Year = c(2011, 2012, 2013, 2014,
2015, 2016, 2017, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2011,
2012, 2013, 2014, 2015, 2016, 2017, 2011, 2012, 2013, 2014, 2015,
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2012, 2013, 2014, 2015, 2016, 2017, 2011, 2012, 2013, 2014, 2015,
2016, 2017, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2011, 2012,
2013, 2014, 2015, 2016, 2017, 2011, 2012, 2013, 2014, 2015, 2016,
2017, 2011, 2012, 2013, 2014, 2015, 2016, 2017), ggroup = c(1510,
1510, 1510, 1510, 1510, 1510, 1510, 1010, 1010, 1010, 1010, 1010,
1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1510, 1510,
1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510,
1510, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1510, 1510, 1510,
1510, 1510, 1510, 1510, 1010, 1010, 1010, 1010, 1010, 1010, 1010,
1510, 1510, 1510, 1510, 1510, 1510, 1510, 1010, 1010, 1010, 1010,
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1010, 1010, 1010, 1010, 1010, 1010, 1510, 1510, 1510, 1510, 1510,
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1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1010, 1010, 1010,
1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010,
1510, 1510, 1510, 1510, 1510, 1510, 1510, 1010, 1010, 1010, 1010,
1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1010, 1510,
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1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510, 1510
), StockPrice.WINS = c(85.19, 84.02, 111.78, 144.23, 130.11,
143.82, 164.08, 56.8, 52.96, 83, 73.82, 48.48, 62.29, 47.47,
90.58, 78.5, 85.94, 62.67, 44.47, 63.47, 42.22, 28.68, 34.92,
50.19, 51.88, 62.66, 70.22, 114.86, 17.86, 22.38, 25.83, 34.09,
36.37, 37.54, 37.85, 106.4, 108.14, 124.91, 112.18, 89.96, 117.7,
125.19, 57.81, 71.9, 104.27, 104.52, 114.38, 117.22, 134.18,
84.76, 86.55, 101.2, 92.45, 77.95, 90.26, 83.64, 43.02, 58.52,
75.46, 57.03, 39.13, 56.56, 94.66, 34.51, 34.69, 50.75, 39.33,
34.04, 54.09, 48.87, 58.36, 56.01, 84.08, 67.42, 53.55, 77.4,
64.64, 23.4, 46.55, 49.69, 37.48, 39.89, 32.76, 51.22, 52.42,
66.54, 85.98, 101.36, 119.64, 117.83, 152.61, 29.6, 39.84, 49.03,
53.58, 37.7, 53.06, 57.94, 29.27, 30.66, 35.3, 28.29, 12.59,
17.31, 16.93, 60.01, 46.44, 23.03, 18.9, 17.99, 34.07, 37.52,
39.57, 43.16, 53.38, 49.05, 40.3, 59.52, 63.58, 93.7, 76.61,
95.1, 80.61, 67.61, 71.23, 73.66, 43.35, 42.75, 62.18, 49.79,
24.66, 57.41, 53.45, 41.75, 67.68, 94.83, 115.58, 98.82, 94.76,
116.82, 72.87, 57.99, 70.65, 69.06, 46.69, 50.14, 54.89, 68.31,
69.3, 90.11, 85.41, 69.75, 83.95, 67.39, 29.76, 51.27, 61.17,
87.68, 86.53, 89.58, 136.68, 39.35, 52.05, 59.42, 65.73, 94.97,
125.15, 128.37, 33.02, 32.74, 38.57, 44.94, 25.7, 31.14, 30.49,
89.48, 106.59, 180.63, 148.85, 125.38, 180.07, 172.85, 36.79,
34.2, 37.74, 23.36, 6.77, 13.19, 18.96, 62, 52.04, 61.87, 61.21,
32, 45.67, 41.4, 21.05, 34.12, 50.4, 49.5, 70.71, 68.32, 91.91,
49.26, 60.4, 83.92, 92.07, 70.79, 101.1, 107.91, 32.14, 39.79,
51.4, 43.86, 40.88, 50.54, 61.59, 106.9, 109.45, 130.03, 129.56,
102.4, 117.19, 154.68, 39.06, 68.05, 80.7, 75.86, 67.51, 75.21,
92.64, 75.41, 94.28, 99.94, 110.32, 136.58, 180.63, 180.63, 51.51,
62.12, 63.39, 60.13, 56.01, 86.08, 127.89, 67.99, 68.35, 79.53,
65.53, 33.49, 37.44, 36.02, 25.24, 38.47, 63.28, 78.05, 63.05,
84.82, 120.55, 50.43, 56.63, 47.27, 45.65, 27.59, 29.33, 25.66,
44.27, 44.53, 55.31, 59.96, 67.33, 78.74, 107.08, 29, 40.63,
46.61, 54.51, 40.81, 31.48, 42.54, 32.49, 57.09, 80.28, 79.39,
86.9, 85.78, 110.32), EPS.WINS = c(5.75, 5.53, 4.74, 4.66, 5.95,
2.92, 13.4368, 5.01, 5.93, 13.4368, 8.11, -10.68, -14.812, -12.93,
11.94, 5.14, 5.65, -14.07, -14.812, -3.71, 3.42, 1.78, 1.65,
2.49, 2.69, 3.01, 3.63, 3.2, 1.34, 1.3, 1.39, 1.7, 1.02, 0.83,
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2.41, 3.23, 4.01, 3.38, 4.2, 5.21, 8.43, 9.7, 7.37, 7.6, 3.85,
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3.13, 4.35, 5.12, 5.21, 5.09, 3.74, 3.1, 1.82, 3.15, 1.3, 2.25,
2.2, 5.19, 4.15, 2.24, 2.49, 4.48, -3.26, -2.53, -6.73, 0.74,
3.65, -4.94, 1.02, 0.43, -1.18, -0.18, 2.45, 1.59, 1.53, 2.23,
1.12, 2.49, 4.12, 8.34, 5.68, 7.34, 0.79, -10.23, -0.75, 1.71,
1.72, 1.75, 1.29, 1.5, 1.17, 1.67, 1.3, 3.58, 3.07, 11.27, 7.6,
5.18, 3.3, 6.21, 9.04, 6.78, 7.44, 5.55, -3.57, -2.9, -0.7, 3.7,
4.13, 5.09, 4.2, 1.65, -1.24, -1.08, 1.42, 2.07, 2.49, 3, 3.81,
4.11, 6.36, -0.55, -0.41, 0.19, 1.56, 1.66, 3.15, 4.54, 0.64,
1.26, 0.63, 2.83, -0.76, -0.57, 2.63, 7.14, 1.56, -1.16, 6.46,
-1.83, -3.35, 4.9, 4.82, 3.2, 2.65, -1.22, -11.27, -3.27, 1.26,
11.28, -0.51, -0.05, 3.93, -14.812, -6.44, 1.71, 3.71, 3.79,
5.02, 6.9, 8.03, 4.97, 9.2, 4.15, 1.07, 4.07, 5.37, -8.29, -1.98,
4.49, 3.7, 6.12, 2.39, 3.09, -5.35, 2.4, 3.89, 5.53, 5.67, 5.94,
5.79, 5.39, 5.25, 4.31, 4.98, 3, 7.56, 5.02, 5.71, 5.8, 9.56,
1.8, 1.84, 2.63, 2.73, 4.32, 6.69, 11.34, 4.82, 3.49, 4.93, 1.69,
3.01, 5.73, 0.49, 4.73, 5.86, 5.46, 5.85, -1.99, -6.41, -0.63,
1.59, 1.7, 4.52, 3.99, 4.47, 4.81, 7.15, 5.64, 4.54, 4.44, 2.75,
2.79, 0.85, -0.31, 3.89, 3.82, 6.93, 4.03, 2.02, 6.21, 6.11,
4.44, 5.79, 4.97, 5.43, 2.97, -1.19, 1.53, 3.75, 4.94, 6.77,
8.01, 9.61, 9.13, 12.23), BookValuePS.WINS = c(27.21, 30.67,
33.58, 34.63, 33.73, 32.72, 46.27, 54.46, 61.75, 75.99, 77.68,
67.77, 45.92, 35.08, 75.5, 80.54, 80.7108, 67.54, 6.79, 16.46,
19.46, 15.53, 12.13, 17.21, 11.53, 10.61, 10.48, 11.88, 3.69,
3.6, 4.11, 3.73, 4.56, 10.85, 11.25, 61.1, 70, 77.78, 80.7108,
80.7108, 77.72, 78.67, 23.92, 20.78, 24.49, 24.38, 23.31, 23.59,
26.31, 31.7, 35.84, 39.36, 40.76, 40.72, 40.12, 44.1, 8.74, 10.75,
11.24, 11.48, 13.95, 14.62, 19.97, 14.38, 17.02, 15.12, 19.18,
18.13, 10.93, 9.57, 30.66, 35.9, 41.81, 45.37, 45.45, 42.23,
38.38, 32.84, 29.49, 29.94, 28, 27.84, 6.58, 30.49, 13.73, 15.4,
17.99, 18.76, 19.77, 20.42, 21.3, 15.32, 14.45, 18.3, 11.89,
9.32, 10.65, 15.79, 24.16, 25.9, 27.44, 30.91, 27.4, 20.71, 13.77,
26.11, 27.77, 20.35, 20.63, 22.18, 20.18, 19.9, 23.58, 24.02,
23.96, 24.3, 23.14, 24.66, 27.31, 46.32, 49.45, 53.63, 44.76,
31.81, 28.14, 26.89, 10.69, 10.33, 11.35, 2.83, 1.6, 0.89, 18.58,
10.33, 13.24, 17.2, 18.73, 18.36, 18.17, 21.7, 47.43, 38.58,
42.32, 41.95, 32.02, 28.08, 25.07, 23.17, 26.13, 29.83, 29.23,
28.36, 30.27, 26.54, 4.89, 5.87, 5.86, 3.45, 3.14, 6.82, 13.25,
29.31, 28.99, 30.23, 31.77, 33.44, 34.33, 37.5, 3.05, 6.98, 7.12,
11.75, 8.21, 6.19, 11.69, 46.95, 46.27, 48.49, 59.59, 56.16,
62.67, 66.32, 16.52, 18.49, 20.89, 17.6, 7.23, 4.59, 5.51, 51.39,
52.67, 50.49, 52.66, 17.11, 11.55, 17.63, 29.17, 32.79, 35.9,
39.31, 41.3, 43.44, 49.75, 48.08, 24.82, 28.53, 32.6, 23.72,
25.26, 28.34, 23.29, 28.62, 30.5, 30.16, 19.76, 20.48, 23.92,
18.16, 20.33, 22.36, 19.22, 15.29, 17.58, 20.82, 13.39, 20.23,
24.65, 23.48, 26.52, 30.77, 37.31, 30.84, 30.8, 33.29, 76.47,
60.74, 65.08, 47.4, 17.58, 20.56, 19.41, 17.27, 29.27, 33.77,
33.13, 41.75, 47.62, 52.18, 48.35, 42.33, 37.07, 37.38, 9.35,
10.06, 13.6, 15.46, 16.71, 18.82, 23.34, 26.1, 28.16, 31.53,
28.61, 26.59, 27.35, 27.41, 8.59, 10.93, 17, 18.18, 15.77, 17.86,
20.94, 13.1, 18.47, 17.24, 16.44, 17.15, 14.38, 15.33, 18.52,
19.31, 21.89, 15.96, 14.06, 14.4, 22.43)), row.names = c(NA,
-287L), class = c("tbl_df", "tbl", "data.frame"))```
CodePudding user response：
You are using different tests, so there is no reason to expect them to produce the same results. Generally it is a bad idea to use a formal statistical test for normality since such tests can be heavily influenced by outliers and sample size. Even a very small departure from normality can be "significant" if the sample size is large enough. Furthermore, many models are extremely robust to departures from normality. A better way to assess normality is to simply inspect a histogram and a QQ plot.
I would highly recommend this Q&A on Cross Validated:
Is normality testing 'essentially useless'?
CodePudding user response：
In short, each of those methods takes a slightly different approach to testing for the normality. Personally, I have tended to use the Shapiro-Wilk test, but each of those options is valid for some subset of data.
The primary question is the nature of the data.
This link gives a good overview of the differences between these: https://www.graphpad.com/guides/prism/latest/statistics/stat_choosing_a_normality_test.htm
Since your data looks like it is not uniquely valued, the Pearson test is probably where you would steer.