This script introduces central concepts of advanced regression analysis by using a data set on stress management and digital health behaviour. The focus lies on understanding how statistical relationships between variables can emerge, change, and be interpreted under different model assumptions.

Using multiple regression models, the script demonstrates key concepts such as direct effects, mediation, moderation, and confounding. In addition, the examples illustrate how theoretical assumptions can be translated into statistical models and how regression results should be interpreted in a substantive context.

Study Background and Analytical Framework

Data Set

The data set contains information on employees’ stress levels, health-related behaviour, and the usage of a digital stress-management app. The simulated data were generated for teaching purposes to demonstrate different types of relationships between variables in regression analysis, including mediation, moderation, and confounding effects.

The central outcome variable is perceived stress. App usage represents the main explanatory variable. Additional variables capture potential behavioural, contextual, and individual factors that may influence the relationship between app usage and stress.

  • App Usage: Frequency and intensity with which employees use a digital stress-management application in their everyday lives.
  • ‘Stress’: Perceived psychological strain and stress level experienced by employees in their work and private life.
  • Healthy Behaviour: Degree to which employees engage in health-promoting activities such as physical exercise, healthy nutrition, sufficient sleep, or relaxation practices.
  • Health Literacy: Ability of employees to access, understand, evaluate, and apply health-related information in everyday decision-making.
  • Workload: Extent of perceived work demands, including time pressure, task intensity, and overall job burden.
  • Interaction: Statistical interaction term representing the combined effect of app usage and workload on stress outcomes.
  • Coaching: Participation in additional professional health or stress-management coaching offered alongside the app.

Data was collected through a survey using multiple multi-item scales. All variables were standardized to improve comparability across measures.

Stress App Usage Health Literacy Healthy Behaviour Workload Coaching
0.7407301 -0.7636647 -0.5814323 -0.6542732 -1.0283159 0.1895842
0.0575681 0.0480959 -0.2483681 -0.1733501 -1.0720491 0.6371499
-0.1110406 0.5711513 1.5554989 -0.5226069 -0.0598663 -0.3993814
-1.0427865 1.0282219 0.0548359 0.3055910 -0.1729671 0.0553653
-2.6357523 0.2282531 0.1141075 1.8763503 -2.5669747 -2.5566299
0.4957400 0.6123542 1.7131650 0.4319743 0.9885448 1.0772031
0.3335088 -1.1469308 0.4485133 -0.2392668 0.2052746 0.0240752
0.7562756 -1.3213169 -1.2919185 -1.0869358 2.3509981 0.2435174
-0.7368779 1.2052789 -0.7088679 1.2922129 0.6365746 -0.4401187
0.9927979 -0.7294461 -0.4656571 -0.5651688 -0.4847351 0.5818479
0.7170371 0.1206094 1.2180700 0.9883189 2.7285296 0.5996346
1.6859359 0.7598453 0.3465642 0.2350294 2.7630307 3.0168688
-1.3638485 0.6164633 0.3878648 -0.6602009 -1.2490931 -0.9913915
-0.8977765 0.6791940 0.0953467 1.0943541 0.4224796 -0.7558462
0.5753093 -0.5734356 -0.5767590 -0.5716075 -0.2512812 -0.2201164
-1.7567410 2.0513990 1.7856148 2.4739064 0.1432005 -0.7961400
-0.0115368 -0.6148584 0.4857568 -0.3648385 0.1833042 -0.8724671
0.8186273 -1.2944183 -1.9993497 0.0061828 -1.2918680 -0.1910724
-0.0996904 -0.1088824 0.6909665 0.1553562 0.2407948 0.8154640
0.3154911 0.2482585 -0.4930138 -0.0400323 1.6904286 -0.1976678
1.1564429 -0.5747214 -1.0930292 0.0585854 -0.2045762 0.5942193
-1.0764081 0.7554607 -0.2360633 0.2608897 -0.2034240 -1.1590455
0.1998347 0.9978148 -1.0508597 1.8438458 1.3431132 0.4797663
-0.2969195 -0.3488581 -0.7512583 -0.1546038 0.8477299 -1.1212635
1.1551774 -0.8235090 -0.6465366 -1.9297076 -1.6747582 0.7990172
2.1090634 -1.7358146 -1.7170816 -1.5803190 0.1843087 0.9986733
1.1010736 0.3321534 0.8285402 -0.8191136 1.5956454 2.1375867
0.4286488 0.0483370 0.1383946 -1.3856140 1.3596575 0.5786622
-0.5941328 0.8724535 -1.1639313 0.6866236 0.3738695 0.1909279
-0.3085624 -0.0952582 1.2480522 0.3603773 0.6722521 -1.1326409
0.0578496 -0.5969987 0.4137727 -0.9320455 -1.2275252 -0.2882714
0.9431565 -0.5104171 -0.3138055 -1.2400996 0.2551975 1.0860567
-1.7312970 1.0506850 0.8863590 0.3845229 -0.9873623 -1.4333485
-1.4904732 1.2269856 0.8692246 -0.1071969 -0.4956879 -1.0899251
-0.3199314 -0.1951370 0.8121985 0.3230266 0.8845808 -1.0176388
-1.7161868 0.5491683 0.6781444 0.7863962 -1.1680584 -2.0552910
0.6182510 0.3265793 0.5422935 -0.7989207 0.2223024 0.6960472
-0.2605922 0.9240590 -0.0786931 0.2769304 0.3821696 -0.2090041
-0.6709282 0.6825979 -0.3247879 1.1942588 0.0123275 -0.0525429
1.0905562 -0.4884824 -0.3999202 -0.4634624 1.7626712 1.8749878
0.9920619 -0.4028117 -0.7167878 -0.2840973 -1.0546101 1.5856291
-0.1139435 0.3956281 -0.2259214 0.5704099 0.5582643 0.2780522
1.6153132 -2.1309131 -1.2922564 -1.1509523 -0.1301369 0.6501499
-1.3624752 1.1478560 2.1708571 1.6407199 -0.3003915 -0.7054536
-1.0761542 1.2411114 1.2018152 1.4251953 0.4175862 -0.9913346
1.1643013 -1.1337531 -1.1487771 -1.7305868 -1.0526820 0.8615443
0.5555914 -0.3005816 -0.4225218 -0.4525732 -1.3429245 0.5559534
-0.4480888 -0.3102846 -0.4868263 -0.0242086 -0.5318014 -1.8836666
0.1639889 -0.0741170 0.7702341 -0.1199885 1.6929143 1.0765521
1.0227077 -0.8447709 -0.1003302 -2.0431702 0.0131721 0.0629606
0.0212369 0.0080134 0.2391770 -0.4847911 0.2862004 -1.2899493
-1.2112375 -0.2906742 -0.0450488 0.4568588 -0.2300860 -1.6134356
-0.4478012 0.6468315 -0.0594924 0.5853973 0.4239263 -0.1338276
-1.7801561 2.1095725 1.3638008 0.5931683 -0.9846194 -0.9540885
0.1219967 -0.1485443 -0.2439247 0.0381897 1.1047576 0.1143480
-0.9790965 1.2977033 1.5129075 0.8566934 0.5370445 -1.0414605
0.6823705 -1.0437415 -1.5779859 -0.3216663 -0.8407842 0.5034868
-1.6492078 0.8626417 0.5732467 2.0207510 0.0119722 -0.8047064
0.5493157 0.2603240 0.1086285 -0.7956479 0.6674097 0.3623351
0.3013030 0.3338792 0.2014871 -0.0792746 0.5103287 0.6773537
0.7772390 1.1412297 0.3665559 0.7546486 1.4256765 1.4044990
0.9429990 -1.1077526 -0.5227931 -2.2084898 -0.6491728 0.4524216
0.3208913 0.9526098 -0.3522608 0.6863462 1.0633840 1.3773967
0.7197880 -0.3675258 -1.0433684 -0.6529949 0.9845580 0.7683068
0.0132855 -0.5365362 -1.0970300 1.0759739 -0.2029847 0.6580142
0.3749936 -0.2486376 0.2898076 -2.0886583 -1.0086342 0.1236435
-1.0280501 0.1618267 0.4357004 1.1000770 -1.1207679 -0.4092679
-0.8115435 0.3392809 0.0371852 0.9160169 0.4113423 -1.0151263
0.4232588 -1.0569527 0.9137281 -0.2899543 -0.1125036 -0.2193166
-0.4618950 0.8381914 2.0509903 0.7188126 1.7199978 -0.2594686
0.4143372 0.1606055 -0.5114063 -0.3415363 0.4879522 0.7057820
0.6520689 -1.7109238 -2.3447701 -0.5282925 -0.4104395 0.0335373
-0.1787627 0.0090359 0.9978982 -0.5676588 -1.0577744 -0.1517477
1.0216724 -0.9584902 -0.7314029 -0.2439418 -0.6188798 0.9561347
0.5336512 -0.2464245 -0.7100333 0.0885102 0.2975446 0.0897223
0.1691676 -0.6469566 1.0178972 0.2351704 -0.4886724 -0.2955948
-0.3706458 0.5214839 -0.3034208 -0.0011970 0.4672447 0.1579096
2.1979266 -2.5154002 -1.2472036 -2.2251235 -0.3731930 1.6994805
-0.0332176 -0.2814019 0.1665589 -0.0031061 -0.1466069 -0.0453437
0.6382741 -0.7142202 -0.1563174 -0.3191498 -0.7655687 0.6927785
-0.1019596 -0.9912993 -0.0104505 0.1769317 1.8447321 -0.2732661
0.0390397 -0.6111350 0.3722440 -0.1255721 0.2873763 -0.0090248
0.7376752 -0.0314519 -0.3900271 0.0777572 0.1863655 0.7279238
-0.8880583 0.3229613 0.6335100 0.7956442 -1.7177102 -0.9703266
-0.1532326 -0.3767384 -0.2385960 -0.6273017 0.6114117 -0.0998619
0.7951594 -1.6416097 0.3182976 -0.4855948 -1.0558741 -0.2156983
-0.2776870 -0.3523784 1.0897616 -0.9231601 -0.9250378 -0.6076450
1.0680885 0.1795786 0.4225630 0.3596519 0.8674837 1.3021416
0.9872836 -0.9069729 -0.3449241 -1.7794841 -0.4904215 -0.1402351
-0.5612381 0.1037864 1.1421655 0.2438174 -1.7736786 -0.4581202
0.7509193 1.4840523 0.9855611 0.0704927 1.7108874 1.9813663
-1.4372539 0.4981559 0.5367266 0.2208157 -2.3966862 -1.4718663
1.0521236 -0.6778022 0.2244681 -1.6440495 0.5252648 1.7411256
0.7287354 -0.7725108 -0.6494275 -0.9813597 0.9654441 0.3283788
-0.5601470 0.6498931 1.3557844 -0.0244118 -0.6669806 -0.0313433
0.7115374 -0.9243838 -0.6215494 -0.3105653 0.3979717 0.5747907
-1.2378493 0.6356216 2.1893880 0.0922199 0.3928645 -0.7396665
-0.3531444 0.9469882 1.5291827 0.1991105 0.9885941 0.2993437
-0.5999093 0.4487251 -0.2539372 1.1364020 0.4374026 -0.5724348
-0.1781802 -0.4705503 -1.0512797 0.7534124 -0.2845958 -0.2107384
0.6412884 -0.6101646 -0.7326188 0.2735333 0.8651571 0.5281877
0.9719988 0.4813411 0.2427721 -0.6168994 0.7508928 1.5227520
-1.4098052 0.4839361 -0.2650207 1.1984293 -0.9696537 -1.0600523
0.6519511 -0.3296943 -0.3667161 -1.0414330 -1.4294242 0.9545556
-0.0184571 -0.2390761 -0.9758509 -0.2170172 0.1166835 -0.1763393
0.1638470 -0.9412896 -0.0616677 -0.2052804 -0.3133958 -0.5403207
0.7917845 -0.6976117 -0.8077406 -0.6473355 -1.0181546 1.1907577
1.2353475 -1.4006044 -1.6981732 -1.1229014 0.0410681 0.6919425
-0.4749751 -0.0811989 -0.3996737 0.0284124 -1.3494074 -0.8113925
-1.3234369 0.8814501 0.9104299 0.9347670 0.1176232 -1.5626488
0.1160930 -0.4622232 -0.5964282 -0.0645372 -0.6609990 -0.0031516
0.6612326 -0.2185196 0.5967928 -1.2858011 0.9059348 0.8362131
0.7823571 -1.7768114 -1.6476947 -1.4162090 2.3592006 -0.6447687
-0.8547234 0.1933911 -0.0722902 0.3361976 -0.9492612 -1.5902975
-0.1613812 1.4263582 0.5074941 1.4423492 1.0054718 0.6775791
-0.0987238 -0.4320199 0.2874125 -0.5197414 0.7751852 -0.7808007
0.0855564 -0.2348826 0.0902983 -0.6271432 -0.1115800 0.6223148
0.7171591 -0.1400042 -0.6623346 -0.9929856 -0.4913581 0.2049567
0.5966402 -0.9836008 -0.8730832 -1.1950828 1.5181555 -0.6266871
0.2231623 -0.5534640 -1.0489684 -0.4807794 -2.0283009 -0.0957459
-0.0890667 0.0430666 0.1023689 -0.3784787 -0.6790904 0.1335515
1.1033378 -1.2239336 -0.9716722 -0.5047733 -1.4651347 0.6855421
0.6755622 0.3576414 -0.5109286 0.5098775 1.3398859 0.9220988
0.0285267 0.5198995 -0.2744998 -0.6655122 -0.2309346 0.4941260
-3.0309164 2.0089541 1.8430406 1.6918163 -0.5617024 -2.8683555
0.9171758 0.6040609 -0.6736726 0.8052904 3.1114782 1.5179841
-0.1009578 -0.2491454 0.2210949 -0.3484967 -0.0916162 -0.6889283
-0.2570859 -0.3165215 0.0623508 -0.2442103 -0.4815559 -0.8968015
0.7617939 -0.4016058 -0.9861747 -0.5726306 0.2549337 0.7142228
0.0504876 0.4650761 -0.0881682 0.1559245 -1.5954552 0.5057983
0.0360428 0.6598288 1.4403854 -0.1683618 0.4435464 -0.0380202
-0.3282527 -0.0440284 0.4390223 0.2521913 -0.1373002 -1.1996183
0.8507415 -0.4633370 0.0253153 -1.1373109 0.4219776 0.2231574
-0.3639848 0.9228786 -0.4422980 0.2980783 -1.0150164 -0.9262274
3.7471328 -3.3211295 -2.0867052 -3.0358078 -1.0552405 2.6342975
-0.5194414 0.5968914 1.1245488 -0.1254802 -0.7288291 -0.0097009
1.3001946 -0.9001751 -1.4891352 -0.8976866 -0.8026895 1.9149240
0.2673166 -0.3690516 0.7298813 -0.5834146 1.2445448 -0.3499000
0.0746329 1.5388480 1.9088286 0.6794851 1.5219566 0.6939576
2.3334046 -2.4967195 -1.4722481 -1.9967802 -0.1974471 1.3015921
-0.7604094 1.2211968 0.6913986 -0.1339749 -0.3975558 -0.1195381
0.9090339 -1.4955575 -0.2806562 -1.0214751 -0.3679445 -0.1696244
1.6385738 -2.7678311 -1.6015731 -1.6162249 0.0265147 0.8425984
1.8173351 -1.5774561 -1.5436153 -1.9951101 0.0539174 1.0679739
1.3337921 -0.8912376 -1.6312113 -1.5754577 0.2451971 1.5545665
-0.1221108 -0.8568877 -0.5516156 0.8379412 -0.7815830 -1.4470508
1.4285349 -1.3797967 -1.4902601 -0.8521464 -0.8808402 0.8255275
-0.3236470 0.7139503 0.6774149 0.5660776 1.1435495 0.1688000
-1.3833603 0.5036705 2.1014335 0.2464783 -0.5854291 -1.6188006
1.2121504 -0.8882924 -1.3140717 -0.3247173 0.2580837 0.6860806
-1.1770151 0.9099899 0.7780729 0.6998454 -0.0984830 -1.2203492
-1.9708916 1.8085703 0.7592197 0.9788372 -0.9907301 -1.2923214
-0.4341670 0.6030809 0.3187217 0.1663234 0.5433403 0.4307836
-0.1055683 0.7336062 -1.0330843 0.3199944 0.1293879 -0.2415859
0.5876616 0.2924000 -0.1367159 0.0116739 1.3443130 0.7925169
0.0264490 -0.1766120 -0.2990065 -0.6164096 0.0742758 -0.3235047
0.7138649 -1.4418757 0.5514414 -0.3510212 -0.3704274 0.7261156
0.8260122 -1.2146186 -0.3918207 -0.8260784 0.2335701 1.2406939
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-0.5466964 0.3364897 -0.3939807 0.6409078 -0.8699433 -0.4441657
-0.4289436 0.4222920 1.0452645 0.5461220 0.4633257 0.3878386
0.0609267 -0.2312239 -1.0742263 -0.2542380 -0.3719875 -0.3027023
0.7099849 -1.6549128 -1.2869714 -0.9488123 -0.1073873 -0.1262750
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-0.3294859 -0.0875816 -0.4366115 -0.1887015 -0.6863381 -0.4036025
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0.3319290 -0.8420437 -0.5040950 -0.8305777 0.7122835 -0.3150618
-1.1526642 0.6372423 0.5049276 0.5848799 -1.3184268 -0.0996048
0.9225961 0.1773974 0.3557915 -0.4616604 1.4261407 0.9840985
0.3289249 -0.1700367 -0.2334472 -0.2337794 0.3394061 0.7443143
0.6605111 -1.5143404 0.0495769 -2.3072795 1.2867270 -0.0778174
0.2558495 -0.5560222 -0.0506155 -0.6957280 -0.9900577 0.6360462
-0.5836187 0.1765964 2.1300138 0.9028933 0.1231249 -0.1481834
1.8426936 -1.7565863 -0.7638074 -2.0406159 -0.1411140 0.8647748
1.4830700 -1.0928445 -1.1214377 -1.5795261 0.7190856 1.2702963
0.3379267 -0.0988731 0.0218419 -0.2107805 -0.6124004 0.2556559
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-1.3382224 0.0680634 -0.4784669 1.7947029 -0.8122290 -1.3305430
0.0809444 0.1351516 -1.0884940 0.6301421 0.0817775 0.0959979
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-0.1845136 -0.4651607 -0.3688415 0.9924190 -0.0852104 -0.3393308
0.4146233 0.3205819 -0.8890241 0.0446873 -0.5055009 -0.0002624
0.5060294 -0.1447590 -0.2545212 -0.2402705 0.5590565 0.4912003
0.2837098 -1.4673977 -0.2150901 -0.9075742 1.1152341 -0.6654105
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-0.4166893 -0.0471126 -0.5197363 -0.4053727 -2.2199127 -0.3047878
0.1553049 0.1546841 0.1999783 -0.1801893 -0.3588316 0.7201737
0.6198491 -0.6002531 -0.3436679 -1.0861285 2.0028630 0.6072191
0.7032987 -0.2147339 0.0791127 -0.0575792 2.1305138 1.3670955
0.4873744 -0.2543694 -0.9191246 0.2061644 0.1130365 0.8657223
0.6290822 -0.6667438 -1.3380419 -0.6342738 -0.8973927 -0.1049692
-1.2480472 2.2760291 1.9976763 1.6900234 0.1218137 0.2342293
-0.8323865 0.2308029 0.5894766 0.7991108 -0.6885782 -1.1662538
0.5553219 -1.5365203 -1.2780132 -1.1314128 1.3968388 0.3328382
0.3552792 -0.2531347 -0.6325471 -0.5479908 -0.8408137 0.7550913
0.1197173 -0.1890292 -1.2116709 -0.1727695 0.3272802 -0.0053159
-1.2025425 3.2447222 2.2009615 1.8411600 0.5718526 0.8957877
-1.3699421 0.4756664 1.3071409 0.4226844 -0.7923105 -1.5072572
1.1527636 -0.4258135 -0.2836285 -0.8955400 0.8013074 1.6122379
-2.5253948 1.0614161 0.5314801 1.7000917 -0.7828221 -3.5603686
0.2842291 0.7670715 -0.4340728 -1.0587688 0.5821428 0.2251878
0.3101628 -0.8275417 -0.4964982 -0.3398768 1.0440956 -0.0375390
-0.2263546 0.0327808 -0.8114700 -0.2896197 -1.0210305 -0.5201712
0.4980853 -0.8284811 -0.6158599 0.6984236 1.0553204 -1.2120562
-0.7781139 0.2931613 1.6484702 0.0680576 -0.5269008 -1.1895001
-0.1866566 0.4707700 -0.0707435 0.6680106 0.2494747 0.8246054
0.0537639 0.1676006 0.1039809 0.3827771 0.1578452 -0.4596571
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-0.2472919 0.3307761 1.2265344 0.5887932 0.2234757 -0.3706806
-1.8556659 1.1135775 -0.5366486 2.1338474 -1.2606999 -1.5851697
0.2489779 -0.0973903 -1.0170819 0.5420236 -0.1768909 0.1355058
-1.1137362 0.7801690 1.6734673 0.7086318 0.7758202 -0.9688155
-1.8551526 0.2861645 -0.4611207 1.3894360 -2.1953517 -2.1152424
0.4499132 -1.4313148 -0.7453843 -0.6648734 -1.5157280 -1.3613262
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General Research Question

To what extent does the usage of a digital stress-management app influence employees’ perceived stress levels?

The main assumption is that higher app usage is associated with lower stress levels. However, this relationship may be influenced by several additional mechanisms and side effects:

  • Workload is expected to act as a moderator. The effectiveness of the app may depend on employees’ workload. For example, the stress-reducing effect of the app may be weaker under very high workload conditions.

  • Health Literacy is expected to act as a confounder. Employees with higher health literacy may both use the app more intensively and generally experience lower stress levels, potentially creating a spurious association between app usage and stress.

  • Healthy Behaviour is expected to act as a mediator. App usage may encourage healthier lifestyles and coping strategies, which in turn may contribute to lower stress levels.

The bivariate model

The bivariate regression model shows a negative association between app usage and perceived stress. The estimated regression coefficient of b = -0.71 indicates that, on average, stress decreases by about 0.71 units for each additional unit of app usage.

This result supports the general research assumption that more intensive usage of the stress-management app is associated with lower stress levels. However, the model only considers the direct relationship between the two variablesm i.e., the main effect of app use on stress, and does not yet account for additional mechanisms.

Adding a second main effect

The multiple regression model including app usage and workload shows that both variables are independently associated with perceived stress levels.

The regression coefficient for app usage (b = -0.73) indicates that higher app usage is associated with lower stress levels, even after controlling for workload. On average, stress decreases by about 0.73 units for each additional unit of app usage.

At the same time, workload shows a positive association with stress (b = 0.3). This means that employees with higher workload tend to report higher stress levels. On average, stress increases by about 0.3 units for each additional unit of workload, independent of app usage.

Compared to the bivariate model, the effect of app usage remains relatively stable, suggesting that workload does not substantially confound the relationship between app usage and stress in this model.

Moderation (Interaction)

The moderation model examines whether the relationship between app usage and stress depends on employees’ workload. In addition to the main effects of app usage and workload, the model therefore includes an interaction term between both variables.

The coefficient for app usage (b = -0.78) indicates that higher app usage is generally associated with lower stress levels. The coefficient for workload (b = 0.29) shows that higher workload is associated with higher stress levels.

Most importantly, the interaction effect between app usage and workload is positive (b = 0.38). This suggests that the stress-reducing effect of the app becomes weaker as workload increases. In other words, the app appears to be more effective under conditions of lower workload, while its beneficial effect on stress is reduced for employees experiencing high workload.

The model therefore provides evidence for a moderation effect: the impact of app usage on stress depends on the level of workload.

While the graphical notation above reflects the substantive idea that moderation describes how the relationship between two variables (app use and stress) depends on a third variable (workload), it does not fully correspond to the statistical model that is actually estimated.

From a statistical perspective, moderation is implemented by creating an interaction term. This interaction term is calculated by multiplying the two explanatory variables (app use × workload) and including this product as an additional predictor in the regression model.

For this reason, moderation can also be represented graphically by an arrow pointing directly from the interaction term to the outcome variable, rather than by an arrow pointing at the relationship itself. In more formal terms, the moderation model therefore takes the following form:

As an alternative illustration of a moderation effect, a so called interaction plot can be generated. The interaction plot visualizes the moderation effect by showing how the relationship between app usage and stress changes across different levels of workload.

The x-axis represents app usage, while the y-axis represents predicted stress levels. Separate regression lines are displayed for different levels of workload. Typically, these levels correspond to low workload (M - 1SD), average workload (M), and high workload (M + 1SD).

If the regression lines are parallel, this would indicate that the effect of app usage on stress is constant across all workload levels and that no moderation effect exists. In contrast, non-parallel lines indicate that the effect of app usage changes depending on workload.

In the present model, the lines differ in their slopes. The negative relationship between app usage and stress is stronger under lower workload conditions, while it becomes weaker as workload increases. This graphical pattern corresponds to the positive interaction coefficient observed in the regression model.

The interaction plot therefore helps to interpret moderation effects substantively by translating the statistical interaction term into conditional relationships between variables.

Confounding

Confounding occurs when the association between an explanatory variable and an outcome variable is partly caused by a third variable that is related to both. In such situations, the observed relationship between two variables may overestimate, underestimate, or even falsely suggest a direct effect.

In the present example, health literacy acts as a confounder because it is associated with both app usage and stress levels. Employees with higher health literacy tend to use the stress-management app more intensively (b = 0.62) and also report lower stress levels (b = -0.23).

After controlling for health literacy, app usage still shows a substantial negative association with stress (b = -0.57). This indicates that higher app usage is associated with lower stress levels even when differences in health literacy are taken into account.

At the same time, the reduction of the app usage coefficient compared to the bivariate model suggests that part of the observed relationship between app usage and stress can indeed be explained by differences in health literacy. The model therefore demonstrates a classical confounding structure in which a third variable influences both the predictor and the outcome.

Mediation

Mediation describes a mechanism in which the effect of an explanatory variable on an outcome variable is transmitted through an additional intermediate variable. In this context, the explanatory variable influences the mediator, which in turn influences the outcome.

In the present example, healthy behaviour acts as a mediator between app usage and stress. Higher app usage is strongly associated with healthier behaviour (b = 0.76). At the same time, healthier behaviour is associated with lower stress levels (b = -0.48).

Even after including the mediator in the model, app usage still shows a direct negative association with stress (b = -0.34). This indicates that the relationship between app usage and stress is only partly explained by healthy behaviour.

Compared to models without the mediator, the direct effect of app usage becomes smaller after controlling for healthy behaviour. This reduction suggests that part of the stress-reducing effect of app usage operates indirectly through healthier lifestyles and behavioural changes promoted by the app.

The model therefore provides evidence for partial mediation: app usage appears to reduce stress both directly and indirectly by encouraging healthier behaviour.

Complete model

The complete model combines direct effects, mediation, moderation, and confounding within a single analytical framework. The model therefore provides a more comprehensive representation of the mechanisms through which app usage is associated with employees’ stress levels.

For didactic reasons, the moderation effect is first illustrated using the intuitive graphical notation in which the moderator points toward the relationship between app usage and stress.

The formally correct statistical representation includes the interaction term (App usage \(\times\) Workload) as an additional predictor pointing directly to the outcome variable.

App usage continues to show a direct negative association with stress (b = -0.27), indicating that higher app usage is generally related to lower stress levels even after controlling for all additional variables and mechanisms included in the model.

At the same time, healthy behaviour acts as a mediator. Higher app usage is strongly associated with healthier behaviour (b = 0.76), while healthier behaviour is associated with lower stress levels (b = -0.46). This suggests that part of the stress-reducing effect of the app operates indirectly through behavioural changes promoted by the app.

The model also includes the moderation effect of workload. Workload itself is positively associated with stress (b = 0.3), meaning that employees experiencing higher workload tend to report higher stress levels. In addition, the positive interaction effect between app usage and workload (b = 0.36) indicates that the beneficial effect of app usage becomes weaker as workload increases.

Furthermore, health literacy acts as a confounder. Employees with higher health literacy tend to use the app more intensively (b = 0.62) and also report lower stress levels (b = -0.25). Controlling for health literacy therefore helps to separate the actual app effect from differences related to employees’ general ability to access and apply health-related information.

Compared to the simple bivariate model, the direct effect of app usage on stress level becomes substantially smaller after controlling for the complex dependency structure. The regression coefficient is now only about one third of the size observed in the original bivariate model, suggesting that a considerable part of the association between app usage and stress is transmitted indirectly through external variables.

Overall, the model demonstrates that the relationship between app usage and stress is not purely direct but emerges through multiple interconnected mechanisms. The results illustrate how mediation, moderation, and confounding can simultaneously shape regression coefficients and their substantive interpretation.