2. Linear Model

The linear model is used to analyze the relationship between two or more variables.

Example 1:

The gain of a transistor is the difference between the emitter and the collector. The Gain variable (in hFE) can be controlled in the ion deposition process through the variables Emission time (in minutes) and Ion dose ($ \times 10^{14} $) variables. The data is shown in Table 1. Our aim is to evaluate the linear relationship between transistor gain and the covariates Emission Time and Ion Dose.

Time	Dose	Gain
195	4	1004
255	4	1636
195	4.6	852
255	4.6	1506
225	4.2	1272
225	4.1	1270
225	4.6	1269
195	4.3	903
255	4.3	1555
225	4	1260
225	4.7	1146
225	4.3	1276
225	4.72	1225
230	4.3	1321

We will upload the data to the system.

Configuring as shown in the figure below to perform the analysis.

Then click Calculate to get the results. You can also generate the analyses and download them in Word format.

The results are:

ANOVA Table

	D.F.	Sum of Squares	Mean Square	F Stat.	P-value
Time	1	630967.86	630967.865	474.40781465	0
Dose	1	20998.23	20998.234	15.78800898	0.0026287995665368
Time:Dose	1	121.00	121.000	0.09097665	0.7691183430521420
Residuals	10	13300.12	1330.012

Expĺoratory Analysis (residues)

Minimum	1Q	Median	Mean	3Q	Maximum
-44.58414	-25.94015	-3.266019	0	24.62883	63.20097

Coefficients

	Estimate	Standard Deviation	T Stat.	P-value
Intercept	71.1733231	1,970.46088	0.03612014	0.971897
Time	8.1533801	8.726180	0.93435849	0.372131
Dose	-289.648874	457.471922	-0.6331511	0.540843
Time:Dose	0.6111111	2.026074	0.30162337	0.769118

Descriptive measure for Goodness-of-Fit

Standard deviation of residuals	Degrees of Freedom	R^2	Adjusted R^2
36.46932	10	0.980011	0.9740149

Confidence Interval for the parameters

	2.5 %	97.5 %
Intercept	-4319.287130	4461.633776
Time	-11.289760	27.596520
Dose	-1308.959837	729.662088
Time:Dose	-3.903262	5.125484

Prediction Interval

Gain	Time	Dose	Fitted Value	Lower Limit	Upper Limit	Standard Deviation
1,004	195	4.00	979.1536	915.3027	1043.0045	28.656620
1636	255	4.00	1615.0231	1551.6616	1678.3846	28.436947
852	195	4.60	876.8643	815.6809	938.0477	27.459417
1506	255	4.60	1534.7338	1473.9136	1595.5539	27.296389
1272	225	4.20	1266.6586	1241.9861	1291.3311	11.073149
1270	225	4.10	1281.8735	1252.1883	1311.5586	13.322842
1269	225	4.60	1205.799	1174.5787	1237.0194	14.011854
903	195	4.30	928.0090	887.9554	968.0625	17.976226
1555	255	4.30	1574.8784	1535.4958	1614.2611	17.675135
1260	225	4.00	1297.0883	1261.0430	1333.1337	16.177353
1146	225	4.70	1190.5841	1152.7665	1228.4018	16.972751
1276	225	4.30	1251.4437	1229.4927	1273.3947	9.851714
1225	225	4.72	1187.5412	1148.3144	1226.7679	17.605162
1321	230	4.30	1305.3495	1282.8141	1327.8849	10.113995

Summary of Residuals Analysis

N.Obs	Time	Dose	Residuals	Studentized Residuals	Standardized Residuals	Leverage	DFFITS	DFBETA	D-COOK
1	195	4.00	24.846387	1.1147927	1.1015027	0.61743967	1.41625490	-0.94199418165	0.4895597961
2	255	4.00	20.976913	0.9108462	0.9187053	0.60800973	1.13439076	0.75939078067	0.3272862194
3	195	4.60	-24.864288	-1.0402662	-1.0360193	0.56692711	-1.19022098	-0.75218753067	0.3512706755
4	255	4.60	-28.733762	-1.2162085	-1.1880774	0.56021533	-1.37266797	0.87352758220	0.4495152217
5	225	4.20	5.341425	0.1460048	0.1537206	0.09219065	0.04652787	-0.00004549144	0.0005999243
6	225	4.10	-11.873462	-0.3338476	-0.3497473	0.13345608	-0.13101533	0.00011212495	0.0047097360
7	225	4.60	63.200975	2.2127130	1.8770621	0.14761681	0.92082173	-0.00056023133	0.1525450911
8	195	4.30	-25.008951	-0.7720700	-0.7881510	0.24296384	-0.43739033	0.02076432661	0.0498406795
9	255	4.30	-19.878424	-0.6029964	-0.6231508	0.23489299	-0.33410935	-0.01574389071	0.0298039755
10	225	4.00	-37.088350	-1.1532997	-1.1347232	0.19677029	-0.57082381	0.00042262343	0.0788568872
11	225	4.70	-44.584138	-1.4566189	-1.3812096	0.21659532	-0.76590938	0.00035872582	0.1318627421
12	225	4.30	24.556312	0.6802984	0.6993418	0.07297401	0.19086997	-0.00019860744	0.0096248792
13	225	4.72	37.458840	1.1981057	1.1728416	0.23303688	0.66041999	-0.00029388986	0.1044886083
14	230	4.30	15.650523	0.4280325	0.4466624	0.07691128	0.12355207	0.00159139594	0.0041557123

Criterion

Diagnostic	Formula	Value
hii (Leverage)	(2*(p+1))/n	0.57
DFFITS	2* Square root ((p+1)/n)	1.1
DCOOK	4/n	0.2857143
DFBETA	2/Square root(n)	0.53
Standardized Residuals	(-3,3)	3
Studentized Residuals	(-3,3)	3

Normality Test

	Statistics	P-value
Anderson-Darling	0.3592592	0.3978779
Shapiro-Wilk	0.9453507	0.4911092
Kolmogorov-Smirnov	0.1614289	0.4142572
Ryan-Joiner	0.9761196	0.5098

Homoscedasticity Test - Breusch Pagan

Statistics	DF	P-value
0.2119295	1	0.6452593

Homoscedasticity Test - Goldfeld Quandt

Variable	Statistics	DF1	DF2	P-value
Time	0.212367391304331	2	1	0.324428433008958
Dose	1.70140146940424	2	1	0.953159041390766

Independence Test - Durbin-Watson

Statistics	P-value
1.71481	0.8201288

Lack of Fit Test


	$\qquad \quad$Lack of fit test
	Input data do not contain replicas

Outliers (X)

Observations	Leverage	Criterin
1	0.6174397	0.5714286
2	0.6080097	0.5714286

Outliers (atypical points)

Observation	T-Value	P-Value	P-Value Bonferroni
7	2.212713	0.05420479	0.758867

Influential Point

Observations	DFFITS	Criterion
1	1.42	± 1.07
2	1.13	± 1.07
3	-1.19	± 1.07
4	-1.37	± 1.07

Influential Points

Observations	DCOOK	Criterio
1	0.4895598	0.2857143
2	0.3272862	0.2857143
3	0.3512707	0.2857143
4	0.4495152	0.2857143

Example 2:

The monthly net salary of employees at a federal university is linearly related to the variables City of employment (Campus the Campinas and Curitiba), Education (Higher Education Completed, Postgraduate Degree Completed or High School Degree Completed), Length of Service at the university, total bonuses in the month and Hours Worked in the month. Data was collected from 22 employees. This data can be analyzed in the table below.

Salary	City	Education	Length of service years	Bonus	Hours worked
2024.84	Campinas	ES	3	99	149.6
2017.15	Campinas	POS	3	95.98	147.4
2014.94	Campinas	EM	3	104.19	146.5
2047.4	Campinas	POS	4	97.52	150.7
2043.59	Campinas	POS	4	105.46	150.1
1913.15	Curitiba	ES	4	94.83	140.2
2064.25	Campinas	POS	5	99.19	150.9
1913.44	Curitiba	ES	5	99.03	141.3
2076.47	Campinas	ES	7	100.82	151.4
2075.83	Campinas	ES	7	97.49	151.3
1950.89	Curitiba	POS	7	107.05	142.8
1915.33	Curitiba	POS	7	108.71	142.3
2096.7	Campinas	ES	11	97.28	153.6
2092.04	Campinas	ES	11	101.96	153.6
1988.75	Curitiba	EM	11	106.77	143.6
1980.92	Curitiba	ES	11	97.13	143.1
2115.13	Campinas	ES	12	93.14	155.3
2101.45	Campinas	ES	12	97.55	154.6
1999.28	Curitiba	ES	12	106.29	145.6
2142.51	Campinas	EM	13	101.89	156
2013.03	Curitiba	EM	14	100.45	145.9
2232.56	Campinas	EM	19	100.63	162.4